document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=87391>Question 87391</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help understaning how to solve these types of problems...can someone show me thanks...\r<BR>\n" );
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document.write( "Problem #1<BR>\n" );
document.write( "A coin is tossed 6 times. Find the probabilities of getting the following.\r<BR>\n" );
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document.write( "a) Exactly 3 heads\r<BR>\n" );
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document.write( "Problem #2<BR>\n" );
document.write( "Quality Control A factory tests a random sample of 20 transistors<BR>\n" );
document.write( "for defective transistors. The probability that a particular<BR>\n" );
document.write( "transistor will be defective has been established by past<BR>\n" );
document.write( "experience as .05.\r<BR>\n" );
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document.write( "a) What is the probability that there are no defective transistors<BR>\n" );
document.write( "in the sample?\r<BR>\n" );
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document.write( "Problem#3<BR>\n" );
document.write( "Flu Inoculations A flu vaccine has a probability of 80%<BR>\n" );
document.write( "of preventing a person who is inoculated from getting the<BR>\n" );
document.write( "flu. A county health office inoculates 83 people. Find the<BR>\n" );
document.write( "probabilities of the following.\r<BR>\n" );
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document.write( "a. Exactly 10 of the people inoculated get the flu.<BR>\n" );
document.write( "b. No more than 4 of the people inoculated get the flu.<BR>\n" );
document.write( "c. None of the people inoculated get the flu.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #63322 by </B><A HREF=/tutors/aboutme.mpl?userid=longjonsilver><B>longjonsilver(2297)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=longjonsilver><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=longjonsilver><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=63322\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> these are all examples of a binomial distribution where you have 2 outcomes: a coin is tossed so HEAD or TAIL, an experiment is done so SUCCESS or FAIL etc.\r<BR>\n" );
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document.write( "If you drew out all the possible outcomes on a probability tree you could find the answers that way, diagramatically but it is tedious for more convoluted questions like yours here. \r<BR>\n" );
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document.write( "All the examples though follow a simple mathematical process:\r<BR>\n" );
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document.write( "Let x = the event we are interested in, so P(x=0) is the probability that x occurs no times. For this to be true, say out of 6 attempts then the \"other\" outcome would occur 6 times.\r<BR>\n" );
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document.write( "Similarly, if x occurred 2 times then the other 4 occurrences must be the \"other\" outcome... the total always adds up to 6, since there are 6 \"attempts\"\r<BR>\n" );
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document.write( "If one outcome is x then other outcome is y and seeing as there are only 2 outcomes possible, then their probabilities have to add up to 1.\r<BR>\n" );
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document.write( "So, all the possible outcomes are:\r<BR>\n" );
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document.write( "P(x=0) = 6C0 * (x)^0 * (y)^6<BR>\n" );
document.write( "P(x=1) = 6C1 * (x)^1 * (y)^5<BR>\n" );
document.write( "P(x=2) = 6C2 * (x)^2 * (y)^4<BR>\n" );
document.write( "P(x=3) = 6C3 * (x)^3 * (y)^3<BR>\n" );
document.write( "P(x=4) = 6C4 * (x)^4 * (y)^2<BR>\n" );
document.write( "P(x=5) = 6C5 * (x)^5 * (y)^1<BR>\n" );
document.write( "P(x=6) = 6C6 * (x)^6 * (y)^0\r<BR>\n" );
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document.write( "Can you see the very simple pattern here... there are 3 parts to each: <BR>\n" );
document.write( "there is the probability of X raised to a power and the probability of y raised to another power and those 2 powers always add up to 6 (in this case).\r<BR>\n" );
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document.write( "The 6C0 etc is the \"combinations\"... you can get this from a formula or on some calculators or from Pascals Triangle.\r<BR>\n" );
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document.write( "eg consider tossing 3 coins. How many heads?\r<BR>\n" );
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document.write( "HHH --> one way...one combination<BR>\n" );
document.write( "mathematically, 3C3 = 1\r<BR>\n" );
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document.write( "HHT HTH THH --> three ways... three combinations<BR>\n" );
document.write( "mathematically, 3C2 = 3\r<BR>\n" );
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document.write( "HTT THT TTH --> three ways... three combinations<BR>\n" );
document.write( "mathematically, 3C1 = 3\r<BR>\n" );
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document.write( "TTT --> one way...one combination<BR>\n" );
document.write( "mathematically, 3C0 = 1\r<BR>\n" );
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document.write( "================================================================<BR>\n" );
document.write( "So, enough of the tutorial:\r<BR>\n" );
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document.write( "1. coin tossed 6 times. P(h=3) where h = number of heads and t = number of tails<BR>\n" );
document.write( "P(h) = 1/2<BR>\n" );
document.write( "P(t) = 1/2\r<BR>\n" );
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document.write( "P(h=3) = 6C3 * P(h)^3 * P(t)^3\r<BR>\n" );
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document.write( "--> 6C3 = (6*5*4)/(3*2*1) = 20\r<BR>\n" );
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document.write( "So, P(h=3) = 6C3 * P(h)^3 * P(t)^3 becomes<BR>\n" );
document.write( "P(h=3) = 20 * (1/2)^3 * (1/2)^3<BR>\n" );
document.write( "P(h=3) = 20 * (1/8) * (1/8)<BR>\n" );
document.write( "P(h=3) = 20 * 1/64<BR>\n" );
document.write( "P(h=3) = 20/64<BR>\n" );
document.write( "P(h=3) = 10/32<BR>\n" );
document.write( "P(h=3) = 5/16\r<BR>\n" );
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document.write( "2. Quality Control A factory tests a random sample of 20 transistors<BR>\n" );
document.write( "for defective transistors. The probability that a particular<BR>\n" );
document.write( "transistor will be defective has been established by past<BR>\n" );
document.write( "experience as .05. <BR>\n" );
document.write( "a) What is the probability that there are no defective transistors<BR>\n" );
document.write( "in the sample? \r<BR>\n" );
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document.write( "2 outcomes are:<BR>\n" );
document.write( "P(Defective) = 0.05<BR>\n" );
document.write( "P(OK) = 0.95\r<BR>\n" );
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document.write( "Let X = number of defective transistors:<BR>\n" );
document.write( "P(X=0) = 20C0 * P(defective)^0 * P(OK)^20<BR>\n" );
document.write( "P(X=0) = 20C0 * (0.05)^0 * (0.95)^20<BR>\n" );
document.write( "P(X=0) = 1 * 1 * (0.95)^20<BR>\n" );
document.write( "P(X=0) = (0.95)^20<BR>\n" );
document.write( "use a calculator to find this.\r<BR>\n" );
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document.write( "3. Flu Inoculations A flu vaccine has a probability of 80%<BR>\n" );
document.write( "of preventing a person who is inoculated from getting the<BR>\n" );
document.write( "flu. A county health office inoculates 83 people. Find the<BR>\n" );
document.write( "probabilities of the following. <BR>\n" );
document.write( "a. Exactly 10 of the people inoculated get the flu.<BR>\n" );
document.write( "b. No more than 4 of the people inoculated get the flu.<BR>\n" );
document.write( "c. None of the people inoculated get the flu.\r<BR>\n" );
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document.write( "Let X = get flu<BR>\n" );
document.write( "so Y = not get flu<BR>\n" );
document.write( "--> i have defined them this way round because the question asks about people NOT getting flu.\r<BR>\n" );
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document.write( "P(X) = 0.20<BR>\n" );
document.write( "P(Y) = 0.80\r<BR>\n" );
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document.write( "a. P(X=10) = 83C10 * P(X)^10 * P(Y)^73<BR>\n" );
document.write( "P(X=10) = 83C10 * (0.2)^10 * (0.8)^73<BR>\n" );
document.write( "whatever this equals using your calculator.\r<BR>\n" );
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document.write( "b. probability that no more than 4 people get the flu means we need to add up each probability for \"no-one\", \"only 1 person gets flu\", \"only 2 people get flu\", \"only 3 people get flu\" and \"only 4 people get flu\"\r<BR>\n" );
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document.write( "So you will have to work out<BR>\n" );
document.write( "P(X=0)<BR>\n" );
document.write( "P(X=1)<BR>\n" );
document.write( "P(X=2)<BR>\n" );
document.write( "P(X=3)<BR>\n" );
document.write( "P(X=4)\r<BR>\n" );
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document.write( "and add these up... tedious, yes :-)\r<BR>\n" );
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document.write( "Good luck.\r<BR>\n" );
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document.write( "and try c. yourself :-)\r<BR>\n" );
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document.write( "cheers<BR>\n" );
document.write( "Jon<BR>\n" );
document.write( " </SPAN>\n" );
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