Lesson Typical probability problems from the archive

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<H2>Typical probability problems from the archive</H2>

<H3>Problem 1</H3>Every Monday, James has a math class and a biology class. &nbsp;The probability that he will skip his math class is &nbsp;0.24, 
while the probability he will skip his biology class is &nbsp;0.11. &nbsp;If the probability he will attend both classes is &nbsp;0.69, 
what is the probability he will skip both classes?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.1123689.html>https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.1123689.html</A>

<B>Solution</B>

<pre>
The probability that he will attend Math class    = 1 - 0.24 = 0.76.

The probability that he will attend Biology class = 1 - 0.11 = 0.89.

The probability that he will attend both classes  = 0.69    ( ! given ! )

The probability that he will attend at least one of the two classes =

0.76 + 0.89 - 0.69 = 0.96.    (*)

The probability that he will skip both classes is the complement to (*)

1 - 0.96 = 0.04.

<U>Answer</U>.  The probability under the question is  0.04.
</pre>

<H3>Problem 2</H3>For safety reasons, &nbsp;5&nbsp; different alarm systems were installed in the vault containing the safety deposit boxes at a Beverly Hills bank. 
Each of the &nbsp;5&nbsp; systems detects theft with a probability of &nbsp;0.8&nbsp; independently of the others.
The bank, &nbsp;obviously, &nbsp;is interested in the probability that when a theft occurs, &nbsp;at least one of the &nbsp;5&nbsp; systems will detect it. 
What is the probability that when a theft occurs, &nbsp;at least one of the &nbsp;5&nbsp; systems will detect it?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1123951.html>https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1123951.html</A>

<B>Solution</B>

<pre>
For each single system of five, the probability that it will not detect a theft is  <U>the complement</U>  to 0.8,  i.e.  1-0.8 = 0.2.

The probability that NO ONE of five systems will detect a theft is  {{{0.2^5}}} = 0.00032.

The probability that when a theft occurs, at least one of the 5 systems will detect it is <U>THE COMPLEMENT</U> to it,  i.e.  1 - 0.00032 = 0.99968.
</pre>

<H3>Problem 3</H3>A box contains &nbsp;44 red, &nbsp;33 white and &nbsp;44 green balls. &nbsp;Two balls are drawn out of the box in succession without replacement. 
What is the probability that both balls are the same color?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1116024.html>https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1116024.html</A>

<B>Solution</B>

<pre>
1.  44 + 33 + 44 = 121.

2.  The probability to get two red balls     = {{{(44*43)/(121*120)}}}.

The probability to get two white balls   = {{{(33*32)/(121*120)}}}.

The probability to get two green balls   = {{{(44*43)/(121*120)}}}.

3.  The probability to get two balls of the same color is

{{{(44*43)/(121*120) + (33*32)/(121*120) + (44*43)/(121*120)}}} = {{{(44*43+33*32+44*43)/(121*120)}}} = {{{(4*43+3*32+4*43)/(11*120)}}} = {{{440/(11*120)}}} = {{{40/120}}} = {{{1/3}}} = 0.3333 = 33.33%.
</pre>

<H3>Problem 4</H3>When two balanced dice are rolled, &nbsp;there are &nbsp;36&nbsp; possible outcomes. &nbsp;Find the probability that the sum is a multiple of &nbsp;3&nbsp; or greater than &nbsp;8.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1116828.html>https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1116828.html</A>

<B>Solution</B>

<pre>
Wanted outcomes are 3, 6, 9, 10, 11, 12.

3 = 1+2, or 2+1,                ====>  so there are 2 ways to get the sum of 3.

6 = 1+5, 2+4, 3+3, 4+2, or 5+1  ====>  so there are 5 ways to get the sum of 6.

9 = 6+3, 5+4, 4+5, 3+6          ====>  so there are 4 ways to get the sum of 9.

10 = 6+4, 5+5, 4+6              ====>  so there are 3 ways to get the sum of 10.

11 = 6+5 and 5+6                ====>  so there are 2 ways to get the sum of 11.

12 = 1+1                        ====>  so there is  1 way  to get the sum of 12.

Thus you have  2 + 5 + 4 + 3 + 2 + 1 = 17 wanted cases of 36 possible outcomes.

The probability under the question is  {{{17/36}}}.
</pre>

<H3>Problem 5</H3>Three airlines serve a small town in Ohio. &nbsp;Airline &nbsp;A&nbsp; has &nbsp;45% &nbsp;of all scheduled flights, &nbsp;airline &nbsp;B&nbsp; has &nbsp;33%&nbsp; and airline &nbsp;C&nbsp; has the remaining &nbsp;22%. 
Their on-time rates are &nbsp;76%, &nbsp;69%, &nbsp;and &nbsp;40%, &nbsp;respectively. &nbsp;A flight just left on-time. &nbsp;What is the probability that it was a flight of airline &nbsp;A?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1119904.html>https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1119904.html</A>

<B>Solution</B>

<pre>
The Probability = the ratio (of the number of the flights A left on-time) to (the number of all flights left on-time) =

{{{(0.45*0.76)/(0.45*0.76 + 0.33*0.69 + 0.22*0.4)}}} = 0.52 = 52%.
</pre>

<H3>Problem 6</H3>Three missiles &nbsp;A, &nbsp;B &nbsp;and &nbsp;C &nbsp;are fired. &nbsp;The probability that &nbsp;A&nbsp; hits the target is &nbsp;0.5, &nbsp;B &nbsp;hits the target is &nbsp;0.7&nbsp; and &nbsp;C&nbsp; hits the target is &nbsp;0.4.
What is the probability the at least one will not hit the target ?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1125218.html>https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1125218.html</A>

<B>Solution</B>

<pre>
The probability that ALL THREE missiles will hit target is the product

0.5*0.7*0.4 = 0.14.

The probability that at least one missile will not hit the target is the COMPLEMENT to it, i.e.  1 - 0.14 = 0.86.
</pre>

<H3>Problem 7</H3>There are seven women and nine men on a committee. &nbsp;A subcommittee of three people is selected at random.  
What is the probability that all three all three people selected are men given that all three all three are the same gender?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1125257.html>https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1125257.html</A>

<B>Solution</B>

<pre>
<U>Answer</U>.  {{{C[9]^3/(C[9]^3 + C[7]^3)}}} = {{{84/(84 + 35))}}} = {{{84/119}}}.
</pre>

<U>Solution</U>

<pre>
How many triples of man can be formed of 9 man ? - {{{C[9]^3}}} = {{{(9*8*7)/(1*2*3)}}} = 84.

How many triples of women can be formed of 7 women ? - {{{C[7]^3}}} = {{{(7*6*5)/(1*2*3)}}} = 35.

Hence, the entire space of events under the question is  84 + 35.

Of them, only 84 are favorable.

It gives you the answer.
</pre>

<H3>Problem 8</H3>Suppose &nbsp;11 &nbsp;manufacturers produce a certain microchip, &nbsp;whose quality varies from manufacturer to manufacturer. 
If you were to select &nbsp;7&nbsp; manufacturers at random,&nbsp; what is the chance that the selection would contain exactly &nbsp;2&nbsp; of the best &nbsp;3 ?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/Permutations/Permutations.faq.question.1125221.html>https://www.algebra.com/algebra/homework/Permutations/Permutations.faq.question.1125221.html</A>

<B>Solution</B>

<pre>
You can select 7 manufactures from 11 manufactures by  {{{C[11]^7}}} ways = {{{(11*10*9*8)/(1*2*3*4)}}} = 330 ways.

It is your sample space, or space of events.

How many of them contain exactly 2 good of the best 3?

First, you can select two good manufactures from the best three by 3 ways.

Second, you can complement these 2 manufactures to 7 by adding 5 manufactures from 11-3 = 8 remaining manufactures,

and you can do it by  {{{C[8]^5}}} ways = {{{(8*7*6)/(1*2*3)}}} = 56 ways.

Thus 3*56 is the number of favorable choices.

Then the probability under the question is  {{{3*56/330}}} = 0.5091 = 50.91%.    <U>ANSWER</U>
</pre>

<H3>Problem 9</H3>The names of four students are placed in a small box and two will be selected. &nbsp;Let &nbsp;1, &nbsp;2, &nbsp;3 &nbsp;and &nbsp;4 &nbsp;denote the students. 
What is the probability that:  &nbsp;&nbsp;&nbsp;&nbsp;a) &nbsp;3 &nbsp;is selected?  &nbsp;&nbsp;&nbsp;&nbsp;b) &nbsp;3 &nbsp;or &nbsp;4 &nbsp;is selected?   &nbsp;&nbsp;&nbsp;&nbsp;c) &nbsp;3 &nbsp;is not selected?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1125508.html>https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1125508.html</A>

<B>Solution</B>

The names of four students are placed in a small box and two will be selected. Let 1, 2, 3 and 4 denote the students. What is the probability that:

a)  &nbsp;&nbsp;<U>3 is selected ?</U>

<pre>
    In all, there are  {{{C[4]^2}}} = {{{(4*3)/2}}} = 6 groups of 2 students, that can be formed of 4 students.

Of these 6 groups, exactly 3 groups contain "3".  They are  (3,1), (3,2), and (3,4).

Thus the probability under the question a) is  {{{3/6}}} = {{{1/2}}}.
</pre>

b)  &nbsp;&nbsp;<U>3 or 4 is selected ?</U>

<pre>
    There is ONLY ONE group of two students, which contain NEITHER "3"  NOR  "4".

This group is (1,2).

The rest 5 of 6 groups necessary contain  EITHER "3"  OR  "4".

Thus the probability under the question b) is  {{{5/6}}}.
</pre>

c)  &nbsp;&nbsp;<U>3 is not selected ?</U>

<pre>
    As we saw in the n. a), there are 3 groups of two students that contain "3".

Hence, of the total possible 6 groups three contain "3", while 3 other do not.

Thus the probability under the question c) is  {{{3/6}}} = {{{1/2}}}.
</pre>

My other lessons on &nbsp;<B>Probability</B>&nbsp; in this site are

<TABLE>
  <TR>
  <TD>&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Simple-and-simplest-probability-problems.lesson>Simple and simplest probability problems</A> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Solving-probability-problems-using-complementary-probability.lesson>Solving probability problems using complementary probability</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Elementary-Probability-problems-related-to-Combinations.lesson>Elementary Probability problems related to combinations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/A-True-False-test.lesson>A True/False test</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/A-multiple-choice-answers-test.lesson>A multiple choice answers test</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Coinciding-birthdays.lesson>Coinciding birthdays</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/A-shipment-containing-fair-and-defective-alarm-clocks-.lesson>A shipment containing fair and defective alarm clocks</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/People-in-a-room-write-down-integer-numbers-at-random.lesson>People in a room write down integer numbers at random</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/A-drawer-contains-a-mixture-of-socks.lesson>A drawer contains a mixture of socks</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Students-studying-foreign-languages.lesson>Students studying foreign languages</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-for-a-computer-to-be-damaged-by-viruses.lesson>Probability for a computer to be damaged by viruses</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Conditional-probability-problems.lesson>Conditional probability problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Using-sample-space-to-solve-Probability-problems.lesson>Using sample space to solve Probability problems</A>
 </TD>
  <TD>
&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Geometric-probability-problems.lesson>Geometric probability problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Advanced-probability-problems-from-the-archive.lesson>Advanced probability problems from the archive</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Challenging-probability-problems.lesson>Challenging probability problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Selected-probability-problems-from-the-archive.lesson>Selected probability problems from the archive</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Experimental-probability-problems.lesson>Experimental probability problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Rolling-a-pair-of-fair-dice.lesson>Rolling a pair of fair dice</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Ten-pairs-of-shoes-are-kept-in-a-rack.lesson>Ten pairs of shoes are kept in a rack</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/A-diplomat-and-spies.lesson>A diplomat and spies</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Elementary-operations-on-sets-help-solving-Probability-problems.lesson>Elementary operations on sets help solving Probability problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Independent-and-mutually-exclusive-events.lesson>Independent and mutually exclusive events</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Unusual-probability-problems.lesson>Unusual probability problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-problem-for-the-Day-of-April-1.lesson>Probability problem for the Day of April, 1</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/OVERVIEW-of-lessons-on-Probability.lesson>OVERVIEW of lessons on Probability</A>

</TD>
 </TR>
</TABLE>

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-II.