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 Probability-and-statistics/195541: 1 solutions Answer 146657 by jim_thompson5910(28598)   on 2009-05-09 14:40:45 (Show Source): You can put this solution on YOUR website!# 1 The probability that the next call will be in person is simply the number of calls answered in person over the number of calls total. So So the chances of a call answered in person are which in decimal form is 0.6 which gives a 60% chance ================================================================================================== # 2 Take note that there are 4 numbers that 5 on a single die namely: 1, 2, 3, and 4 So the relative frequency is which is 66.67%. So this means that the answer is A) ================================================================================================== # 3 First, we need to find the total number of marbles: 4 yellow + 12 green + 11 black = 27 total So there are 27 marbles in total. To find the chances of NOT yellow, we can do it two ways: Method # 1: Simply add up the number of marbles that are NOT yellow. So add up the 12 green and 11 black to get 23 marbles that are NOT yellow. Now divide it by the total 27 to get: So the chances of selecting a non yellow marble is which is 0.8519 in decimal form and gives a 85.19% chance of occurring ---------------------------- Method # 2: If we pick a marble, it is either yellow or NOT yellow (but not both). There are no other choices. So this means that if we find the chances of picking a yellow marble, we can subtract this from 1 to find the chances of getting a non yellow marble. So.... So the chances of selecting a non yellow marble is again which is 0.8519 in decimal form and gives a 85.19% chance of occurring So we get the same answer using either method. This means that you can use whatever you find easier (to compute or understand) ================================================================================================== # 4 If we look through a deck of cards, we'll find that there are 4 jacks (one of each suit) So So there is a chance of picking a jack. This in decimal form is giving a 7.69% chance. ================================================================================================== # 5 Since the odds against Thunderbolt winning are 11:3, this means that the corresponding odds are Winning: 3 Losing: 11 giving a total of 11+3=14 So ... This means that the probability of winning is giving a decimal value of 0.2143 which means that there's a 21.43% of Thunderbolt winning. So the answer is B) ================================================================================================== # 6 a) First, list out the individual probabilities for every possible scenario: P($300 Prize) = 1/500 P($200 Prize) = 1/500 P($50 Prize) = 3/500 P(Losing) = 996/500 = 249/125 Now simply multiply the probabilities listed above with their corresponding winning value and then add them all up like so: Expected Value = P($300 Prize)*(Value) + P($200 Prize)*(Value) + P($50 Prize)*(Value) + P(Losing)*(Value) Expected Value = (1/500)*(300) + (1/500)*(200) + (3/500)*(50) + (496/500)*(0) Expected Value = 300/500 + 200/500 + 150/500 + 0 Expected Value = 3/5 + 2/5 + 3/10 Expected Value = 6/10 + 4/10 + 3/10 Expected Value = (6 + 4 + 3)/10 Expected Value = 13/10 Expected Value = 1.30 So he expects to earn $1.30 on average (without factoring in the cost) However, we need to subtract the cost of playing a single game to get 1.30 - 2 = -0.70 So he expects to lose 70 cents ---------------------------------------------------------- b) Since the expected winnings is$1.30, this means that the cost must equal the expected winnings to make the expected value equal to zero (since 1.30-1.30=0) So the fair price for the ticket is $1.30 ================================================================================================== # 7 Let r = red ball b = blue ball g = green ball y = yellow ball Let's list out ALL the possible points in the sample space {r,b} {r,g} {r,y} {b,r} {b,g} {b,y} {g,r} {g,b} {g,y} {y,r} {y,b} {y,g} So there are 12 different possible outcomes which means that there are 12 points in the sample space. Take note that first the first ball there are 4 choices and then once we select a ball (and don't put it back) we'll have 3 choices. Multiply these values together to get 4 * 3 = 12 So we get the same answer. ================================================================================================== # 8 We could list out all the possible license plates, but that would take too long and there's a very high chance that we'll repeat values and/or forget to list some elements. Since the first slot must be a vowel, this means that there are 5 choices for the first position. For the second slot, there are 25 choices (since repetition of letters is not allowed). Similarly, there are 24 choices for the third slot. For the digits, there are 10 choices (for numbers 0 - 9). Since repetition is allowed, there are 10 choices for the last digit also. Now simply multiply all of the choices to get 5 * 25 * 24 * 10 * 10 = 125 * 240 * 10 = 300,000 So there are 300,000 different license plates. ================================================================================================== # 9 Simply use the counting principle to find the number of possible outfits: Number of Pants * Number of Shirts * Number of Ties = 7 * 9 * 4 = 63 * 4 = 252 So there are 252 different outfits ================================================================================================== # 10 a) Number of Genders * Number of Colors * Number of Seat Choices = 2 * 3 * 3 = 6 * 3 = 18 So there are 18 different bike configurations. b) Here's the tree: c) Here are ALL of the possible configurations which form the sample space: {male, red, soft} {male, red, medium} {male, red, hard} {male, white, soft} {male, white, medium} {male, white, hard} {male, blue, soft} {male, blue, medium} {male, blue, hard} {female, red, soft} {female, red, medium} {female, red, hard} {female, white, soft} {female, white, medium} {female, white, hard} {female, blue, soft} {female, blue, medium} {female, blue, hard} Note: an example point {female, red, soft} means that the bike has a female frame, is red, and has a soft seat. ================================================================================================== # 11 Let's use the table to answer these questions: a) So the chances are giving a decimal value of 0.4646 or a 46.46% chance b) So the chances are giving a decimal value of 0.4706 or a 47.06% chance c) So the probability is giving a decimal value of 0.5259 or a 52.59% chance ================================================================================================== # 12 For the first course, there are 6 choices. For the second there are 5 choices. For the third, there are 4 etc.. So multiplying all of this together, we get: This means that there are 720 different ways for 6 instructors to teach six sections of a course in mathematics. Note: the factorial button (!) is used as a shortcut. So ================================================================================================== # 13 a) For first place, there are 7 choices. For second place, there are 6 choices (since it's not possible to place in two different positions). For third place, there are 5 choices, etc.. all the way down to 1. So multiplying all of this together, we get This means that there are 5,040 different arrangements. b) For first place, there are 7 choices. For second place, there are 6 choices and finally for third place there are 5 choices. Multiply these values together to get: 7*6*5 = 42*5 = 210 So there are 210 different ways to place in first, second and third place ================================================================================================== # 14 We could use the counting principle as we've done for earlier problems, but we'll have overlap and the sample space is far too large. So let's do it this way: In this case, order does NOT matter since the candidates have no rank over one another (ie one isn't president or secretary). Since order does not matter, we must use the combination formula: Start with the combination formula. Plug in and Subtract to get 9 Expand 15! Expand 9! Cancel Simplify Expand 6! Multiply 15*14*13*12*11*10 to get 3,603,600 Multiply 6*5*4*3*2*1 to get 720 Reduce. So 15 choose 6 (where order doesn't matter) yields 5,005 unique combinations This means that there are 5,005 different ways to select a group of 6 college candidates from a group of 15 applicants for an interview (where the order of the candidates doesn't matter).  Equations/195523: 5x-1/2-x-2/5=8x+11/61 solutions Answer 146650 by jim_thompson5910(28598) on 2009-05-09 13:25:48 (Show Source): You can put this solution on YOUR website!I'm assuming that the equation is ???? Start with the given equation. Multiply EVERY term by the LCD to clear any fractions. Multiply. Combine like terms on the left side. Add to both sides. Subtract from both sides. Combine like terms on the left side. Combine like terms on the right side. Divide both sides by to isolate . Reduce. ---------------------------------------------------------------------- Answer: So the answer is which approximates to .  Equations/195528: Please help me with solving this problem. 3 * 4 1/2 * n = 2 Thank you. 1 solutions Answer 146649 by jim_thompson5910(28598) on 2009-05-09 13:24:12 (Show Source): You can put this solution on YOUR website!First, let's convert into an improper fraction So the mixed fraction converts to the improper fraction This means that then becomes Start with the given equation. Multiply Multiply both sides by 2 Multiply. Divide both sides by to isolate . ---------------------------------------------------------------------- Answer: So the answer is which approximates to . Polynomials-and-rational-expressions/195512: HOW DO I FACTOR PROBLEMS LIKE X^2-9XY+14Y^2 OR 6a^2-5ab-4b^2 please show the work for it thanks 1 solutions Answer 146648 by jim_thompson5910(28598) on 2009-05-09 13:21:03 (Show Source): You can put this solution on YOUR website! # 1 Looking at we can see that the first term is and the last term is where the coefficients are 1 and 14 respectively. Now multiply the first coefficient 1 and the last coefficient 14 to get 14. Now what two numbers multiply to 14 and add to the middle coefficient -9? Let's list all of the factors of 14: Factors of 14: 1,2,7,14 -1,-2,-7,-14 ...List the negative factors as well. This will allow us to find all possible combinations These factors pair up and multiply to 14 1*14 2*7 (-1)*(-14) (-2)*(-7) note: remember two negative numbers multiplied together make a positive number Now which of these pairs add to -9? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -9 First NumberSecond NumberSum 1141+14=15 272+7=9 -1-14-1+(-14)=-15 -2-7-2+(-7)=-9 From this list we can see that -2 and -7 add up to -9 and multiply to 14 Now looking at the expression , replace with (notice combines to . So it is equivalent to ) Now let's factor by grouping: Group like terms Factor out the GCF of out of the first group. Factor out the GCF of out of the second group Since we have a common term of , we can combine like terms So factors to So this also means that factors to (since is equivalent to ) ------------------------------------------------------------ Answer: So factors to # 2 Looking at we can see that the first term is and the last term is where the coefficients are 6 and -4 respectively. Now multiply the first coefficient 6 and the last coefficient -4 to get -24. Now what two numbers multiply to -24 and add to the middle coefficient -5? Let's list all of the factors of -24: Factors of -24: 1,2,3,4,6,8,12,24 -1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. This will allow us to find all possible combinations These factors pair up and multiply to -24 (1)*(-24) (2)*(-12) (3)*(-8) (4)*(-6) (-1)*(24) (-2)*(12) (-3)*(8) (-4)*(6) note: remember, the product of a negative and a positive number is a negative number Now which of these pairs add to -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5 First NumberSecond NumberSum 1-241+(-24)=-23 2-122+(-12)=-10 3-83+(-8)=-5 4-64+(-6)=-2 -124-1+24=23 -212-2+12=10 -38-3+8=5 -46-4+6=2 From this list we can see that 3 and -8 add up to -5 and multiply to -24 Now looking at the expression , replace with (notice adds up to . So it is equivalent to ) Now let's factor by grouping: Group like terms Factor out the GCF of out of the first group. Factor out the GCF of out of the second group Since we have a common term of , we can combine like terms So factors to So this also means that factors to (since is equivalent to ) ------------------------------------------------------------ Answer: So factors to  Signed-numbers/195531: ( sqrt 5 - sqrt7)^2 answer:12 - 2 sqrt 35 ok so i tried the foil method and i got sqrt 5 sqrt7 as my answer can anyone tell me how my book got 12-2 sqrt 35 as an answer 1 solutions Answer 146646 by jim_thompson5910(28598) on 2009-05-09 13:17:38 (Show Source): You can put this solution on YOUR website! Start with the given expression. Expand. Remember something like . Now let's FOIL the expression. Remember, when you FOIL an expression, you follow this procedure: Multiply the First terms:. Multiply the Outer terms:. Multiply the Inner terms:. Multiply the Last terms:. --------------------------------------------------- So we have the terms: , , , and Now add every term listed above to make a single expression. Now combine like terms. So FOILs to . In other words, .  Complex_Numbers/195529: I'm trying to solve the following problem, but get stuck half way through it, can you please help me? 16 + 4i divided by 16 - 4i I've been trying the following, I tried multiply both the numerator and denominator by the conjugate of the denominator and this is as far as I get before getting stuck (16+4i) (16+4i) using the foil method I get 256 +64i+64i+16i^2. When I do the same for the denominator (16-4i)(16+4i) using the foil method I get 272. I combine the numerator and get the following 240+128i and then this is where I'm stuck I don't know what to do after this 240+128i/272. Can you please help me?1 solutions Answer 146645 by jim_thompson5910(28598) on 2009-05-09 13:13:51 (Show Source): You can put this solution on YOUR website! Start with the given expression. Multiply the fraction by . Combine the fractions. FOIL the numerator. FOIL the denominator. Multiply. Combine like terms. Break up the fraction. Reduce. So . So the expression is now in standard form where and  Exponents-negative-and-fractional/195530: Can you show me step by step how to solve this? 8^-5*8^7 ________ 8^2 1 solutions Answer 146644 by jim_thompson5910(28598) on 2009-05-09 13:12:06 (Show Source): You can put this solution on YOUR website!First, simplify the numerator by adding the exponents: So Start with the given expression. Subtract the exponents Subtract Raise 8 to the zeroth power to get 1 So which means that  Graphs/195473: I need help writing an inequality for this problem. I have x amount of dollars to spend on a landscaping project. I want to spend more than 6,000 but less than 10,000. How would I write an inequality to show all of this?1 solutions Answer 146601 by jim_thompson5910(28598) on 2009-05-08 15:45:05 (Show Source): You can put this solution on YOUR website!"I want to spend more than 6,000 " translates to and "less than 10,000" translates to Now combine the inequalities to form  Square-cubic-other-roots/195463: I really do not understand these problems please help Rewrite the following in simlified radical form 1)square root 54x^4 Simplify the following expressions as much as possible 2)6w square root63u^3-usquare root7uw^2 Rationalize the denominator and simplify 3)square root21/square root 771 solutions Answer 146600 by jim_thompson5910(28598) on 2009-05-08 15:43:10 (Show Source): You can put this solution on YOUR website!# 1 Start with the given expression Factor 54 into . Note: 9 is the largest perfect square factor of 54 Factor into Break up the square root. Take the square root of 9 to get 3 Take the square root of to get "x" Multiply So where # 2 Start with the given expression Simplify to get Simplify to get Multiply Factor out the GCF Combine like terms. So where every variable is non negative. # 3 Start with the given expression Multiply both the numerator and denominator by Multiply to get Multiply to get Reduce So  Graphs/195401: Please i need help graphing this! 12x-3y=24 thank you!1 solutions Answer 146543 by jim_thompson5910(28598) on 2009-05-07 23:38:25 (Show Source): You can put this solution on YOUR website! Start with the given equation. Subtract from both sides. Rearrange the terms. Divide both sides by to isolate y. Break up the fraction. Reduce. Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis So we have one point Now since the slope is comprised of the "rise" over the "run" this means Also, because the slope is , this means: which shows us that the rise is 4 and the run is 1. This means that to go from point to point, we can go up 4 and over 1 So starting at , go up 4 units and to the right 1 unit to get to the next point Now draw a line through these points to graph So this is the graph of through the points and  Radicals/195393: Solve sqrt((1/3)x+2)=81 solutions Answer 146542 by jim_thompson5910(28598) on 2009-05-07 23:03:21 (Show Source): You can put this solution on YOUR website! Start with the given equation. Square both sides Square the square root of the left side to eliminate it. Square 8 to get 64 Multiply EVERY term by the LCD to clear the fractions. Distribute and multiply. Subtract from both sides. Combine like terms on the right side. ---------------------------------------------------------------------- Answer: So the answer is  Trigonometry-basics/195285: population growth rates and compounding interest is an example of?1 solutions Answer 146491 by jim_thompson5910(28598) on 2009-05-07 15:39:29 (Show Source): You can put this solution on YOUR website!population growth rates and compounding interest is an example of exponential growth  Sequences-and-series/195291: write amn explicit formula for the following infinite sequence 14,17,20,23.....1 solutions Answer 146489 by jim_thompson5910(28598) on 2009-05-07 15:38:28 (Show Source): You can put this solution on YOUR website!Take note that the first term is 14 and we're adding 3 to each term to get the next term. So this tells us that and So the general formula (which is an arithmetic sequence) becomes where "n" starts at 0 Note: if you want to start at n=1, then simply subtract 3 from 14 (to scale back the terms) to get where "n" starts at 1.  Radicals/195309: Find 2 other pairs of positive integers a and b such that the square root of a * the square root of b = 6*the square root of 51 solutions Answer 146487 by jim_thompson5910(28598) on 2009-05-07 15:35:12 (Show Source): You can put this solution on YOUR website! Start with the given equation Rewrite 6 as Combine the roots. Multiply Square both sides. Take note that So all you need to do is find two numbers that multiply to 180. I'll let you do that.  Equations/195301: x+3+2x=x+31 solutions Answer 146481 by jim_thompson5910(28598) on 2009-05-07 15:21:52 (Show Source): You can put this solution on YOUR website! Start with the given equation. Combine like terms on the left side. Subtract from both sides. Subtract from both sides. Combine like terms on the left side. Combine like terms on the right side. Divide both sides by to isolate . Reduce. ---------------------------------------------------------------------- Answer: So the answer is  Probability-and-statistics/195296: 1 solutions Answer 146473 by jim_thompson5910(28598) on 2009-05-07 12:47:48 (Show Source): You can put this solution on YOUR website!# 1 a) ... Start with the given probability ... Plug in the given values. ... Reduce. ... Use a calculator (or long division) to find the approximate answer. So the chances of a tulip blooming are 0.8378 which is 83.78% (multiply by 100 to get a percentage). ----------------------------------------------------- b) If we multiply the number planted by the probability that a flower will bloom, we will get an approximate number that will bloom. This is because the probability allows us to estimate a population. So let x = number originally planted and y = number of blooming flowers wanted. So the equation is Start with the given equation Plug in (the probability we found earlier) and (the number of tulips wanted) Divide both sides by 0.8378 Divide Rearrange the equation. Round up to the nearest whole number (this guarantees that we pass our target). So if the gardener plants 114 or more bulbs, then at least 95 bulbs should bloom. ===================================================================================================== # 2 Notation note: The notation n(...) means "the number of ...". So n(assualts) is read as "the number of assualts" and n(violent crimes) is read as "the number of violent crimes". To find the chances of an assualt, simply divide the number of recorded assualts (17) by the total number violent crimes (50) like so: So there is a 34% chance that the next violent crime will be an assualt. Note: simply multiply the decimal value 0.34 by 100 to get 34% ===================================================================================================== # 3 In this problem, the population of Pennsylvannia is extra information. In other words, you don't use this info to find the answer (and it's there to confuse you). Number of people who do NOT live in California = Number people in US - Number of people who live in California Number of people who do NOT live in California = 293,655,404 - 35,893,799 = 257,761,605 So Number of people who do NOT live in California = 257,761,605 "What is the probability that a randomly selected US resident did not live in California? " So the probability of selecting a US resident that does NOT live in California is... P(NOT California) = # who do NOT live in California/# total P(NOT California) = 257,761,605/293,655,404 P(NOT California) = 0.877769 So the percentage is 87.7769% which rounds to 87.777% (to the nearest thousandth of a percent). ===================================================================================================== # 4 When they report the odds of 9:4, this means that the chances of losing is and the chances of winning are (note: which is the total). These odds form the ratio and if we plug the given values in, we get the ratio: Start with the given ratio Multiply both sides by 8 Multiply Simplify So our answer is which means that Sally wins$18 (which makes the answer choice C) ===================================================================================================== # 5 First, list out the probabilities for each possible event: P(grand prize) = 1/1000 P(consolation prize) = 2/1000 = 1/500 P(losing) = 997/1000 Note: since there are 3 prize positions, there are 1000-3=997 losing positions. Now simply multiply the probabilities with their corresponding winning value and then add up those products. In other words.... Expected Value = P(Grand Prize)*Value of Grand Prize + P(Consolation Prize)*Value of Consolation Prize + P(losing)*Value of losing Now plug in the given values and evaluate: Expected Value = (1/1000)*(100) + (1/500)*(75) + (997/1000)*(0) Expected Value = 1/10 + 3/20 + 0 Expected Value = 5/20 Expected Value = 1/4 Expected Value = 0.25 So he expects to make $0.25. In other words, on an average try, he'll make 25 cents. However, he spent$3 to play the game so this means that we need to subtract the cost to play from the expected winnings to get: Expected Average Winnings - Cost to Play = $0.25 -$3 = -$2.75 So he expects to lose$2.75 overall which means that the answer is A). ===================================================================================================== # 6 To find the expected value, simply perform a weighted average: Weighted Average: where S = Number of Suspense Movies s = Number of Suspense Movie Viewers C = Number of Comedy Movies c = Number of Comedy Movie Viewers D = Number of Drama Movies d = Number of Drama Movie Viewers So plug in the given values and evaluate: So the expected number of viewers is 5,025 which means that the answer is D) ===================================================================================================== # 7 Take note that the even and odd digits are Even: 0, 2, 4, 6, 8 Odd: 1, 3, 5, 7, 9 So there are 5 even and 5 odd numbers. Also, the number of values from 0 to 9 is 10. So to find the number of combinations, simply multiply the number of choices for each digit. So multiply the number of choices for the first digit (5) by the number of choices for the second digit (10) by the number of choices for the third digit (5): 5*10*5=50*5=250 So 250 different numbers can be formed based on the given criteria. --------------------------------- Side Note (ignore if you understand method given above): if you aren't sure about the multiplication, try using smaller numbers. So let's say we can only use the numbers 1, 2, and 4. How many 2 digit numbers can we form (with repeated values)? Well, we can list them out: 11, 12, 14, 21, 22, 24, 41, 42, 44 From this list, we can see that there are 9 possible numbers. Take note that we have 3 choices for each digit. So multiplying 3*3 gets us 9 which is the original answer. Also, you can draw a tree to visually reinforce the idea. ===================================================================================================== # 8 Whenever we want to find the probability of event A OR event B happening, we simply add the two probabilities like so: ... Start with the given probability ... Expand ... Plug in the given values. ... Add the fractions. ... Add ... Reduce So the chances are which is 0.8 in decimal form which gives an 80% chance. ===================================================================================================== # 9 a) Using the tree, we get the sample space: {b,b,b,b} {b,b,b,g} {b,b,g,b} {b,b,g,g} {b,g,b,b} {b,g,b,g} {b,g,g,b} {b,g,g,g} {g,b,b,b} {g,b,b,g} {g,b,g,b} {g,b,g,g} {g,g,b,b} {g,g,b,g} {g,g,g,b} {g,g,g,g} So for instance {b,g,g,b} means that the couple had a boy, girl, girl, and then a boy (in that order). Note: recall, the sample space is the set of ALL possible outcomes. ------------------------------- b) Since we want to know the chances of the couple having "at least three boys", this means they want to know the chances of having 3 boys OR 4 boys (since at least means that figure or more). Looking back at the list of all possible outcomes (ie the sample space) from part a), we see that we have the combinations for 3 boys: {b,b,b,g}, {b,b,g,b}, {b,g,b,b}, and {g,b,b,b} So there are 4 cases where the couple would have 3 boys. So ----------------------------- Also, since there is only ONE way to have 4 boys (of a total of 4 children), this means that ------------------------------------------------------- Now simply add the two probabilities to find the chances of either one occurring: So the probability of the couple having AT LEAST 3 boys is which is 0.3125 in decimal form which gives a 31.25% chance ===================================================================================================== # 10 a) Since there are 12 face cards (there are 4 groups of jack, queen, and king cards) and there are 52 total cards, this means that there are cards that are NOT face cards. Since we know that the given card is NOT a face card, this means we can focus solely on the 40 non face cards. So the number of the sample space is 40. Also, since there are 4 cards labeled "7", this means that the number of elements in the event space is 4. So So the chances of drawing a 7 given that the card is NOT a face card is which is 0.1 or 10% ------------------------------------------------------------- b) Since spade cards are black and the given card is red, this means that the probability of choosing a spade given the card is red is 0%. In other words, it is IMPOSSIBLE to choose a spade card given the card is red. More formally, we can say that Number of spades that are red = 0 Number of Red Cards = 26 P(spade | red) = 0/26 = 0 So either way, we get the same answer of 0