5.11 Supplemental Exploration A: Conditional probability and independence College admissions college A, a 0.40 probability of being accepted by college B, and a 0.35 probability of being accepted by both will investigate how the probability that you will be accepted by one of these colleges changes if you find out whether you have been accepted by the other. Question 1. counts provided in the following table correspond to the given 0.80. 0.40, and 0.35 probabilities. Fall in the missing counts in the table. Accepted by B Not accepted by B Total Accepted by A 350 800 Not accepted by A Total 400 1000 ANTICIPATION | 5.11.1: Question 2 Among the 800 applicants who were accepted by college A, what proportion were also accepted by college B? 0 4375 Submit Keep your answer in mind as you continue. We will refer to the proportion from #2 as the conditional probability of being accepted by college B, given that a applicant was accepted by college A Definition The conditional probability of B given A is represented P( B | A) and is the probability that B occurs it we know that A has occurred RAPTUCIPATION |5.11.2: Question 3. How does the conditional probability of acceptance by college B, given acceptance by colege words, does learning that an applicant has been accepted by college A make likely (or just as likely) that the applicant will be accepted by colege B? The unconditional probability (P(B) = 940 Keep your answer in mind as you continue. Key idea The conditional probability of event B, given that event A has occurred, is calculated as P(B | A) = PAL PARTICIPATION 5.11.3: Question 4 Use the formula in the Key Idea box to calculate P(BIA). Verify that the answer agrees with what you found earlier. college A is 0:4375. This agrees with the counts in the table. e been accepted by Submit Keep your answer in m ANTICIPATION |5.11.4: Question 5. Suppose that you learn that you have been accepted by at least one of these colleges. You want to calculate P((A and B) | (A or B) ). Use the definition of conditional probability, applied to these compound events.) The probability of being accepted at either Colege X or College Y is 0.76. or 76% of the colleges, not 0.76%, that there is a 76% chance of being accepted by at least one Submit Keep your answer in mind Independence Now suppose that a friend has applied to respectively, and a 0.40 probability of acceptance to both. Question Produce a 2 x 2 table of coun probabilities, Definition Two events A and B are said to be independent if P( B | A) = P(B). in other words, knowing that independent. they are said to be dependent in other words, knowing that event A has occurred does hange the probability that event B will occur. Note that if events A and B are independent, then events not A and not B are also independent, as are A and not B, and so on. ANTICIPATION |5.11.5. Question 7. Calculate the conditional probability that your friend is accepted to college C if they hear that they have been accepted by college D. What is the probability the D if they hear that they have been accepted by college C? Submit ACTIVEVATION 5.11.6. Question & How do these conditional probabilities compare to the (unconditional) probabilities? Does learning about acceptance by one college change the probability of acceptance to the other? Submit ACTIVITY ON 5.11.7: Question 9. be by college D? Justify your answer ACTIVELYATION 5.11.8: Question 10. Is acceptance by college A indepen ance by college 87 Justify your answer. Submit Multiplication Rule conditional probability peacemarge montanaaddit was mary soon probability of acceptance by 6. an simply by rearranging the equation for conditional probability. Definition The general multiplication rule allows you to calculate the probability that the first P( An B) = P(A) x P(B | A) = P(B) x P(A | B). probability of the other event given Note: If A and B are independe this rufe simplifies to the multiplication rule for independent events. P(An B) = P(A) x P(B). ACTIVITY ON 5.11.9: Question 11. bability of acceptance by both G and