Suppose that you have applied to two colleges and believe that you have a 0.80 probability of being accepted by college A, a 0.40 probability of being accepted by college B, and a 0.35 probability of being accepted by both. You 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.

5.11 Supplemental Exploration A: Conditional probability and independence College admissions Suppose that you have applied to two colleges and believe that you have a 0.80 probability of being accepted b centers suppressonly offend accepted by college s and a diss probability of being accepted by both. thether you have been accepted by the other. Question 1 start by coffeefund the giveed prebabies into a zu a table of counts to help make things more concrete. Let's counts provided in the following table correspond to the given 0,80, 0.40, and 0.35 probabilities. Fill in the missing counts in the table. Accepted by B Not accepted by B Total Accepted by A 350 Not accepted by A Total 400 1000 PARTICIPATION 5.11.1: Question 2 Among the 800 applicants who were accepted by college A. what proportion were also accepted by college B? We will refer to the proportion from #2 as the condit pplicant was accepted by college A Definition The conditional probability of B given A is represented we know that A has occurred ANTICIPATION |5.11.2: Question 3. How does the conditional probability of acceptance by college B, given acceptance by college words, does learning that an applicant has been accepted by college A make it more or less likely (or just as likely) that the applicant will be accepted by college B? Submit Key idea red, is calculated as P(B | A) = PAY ANTICIPATION 5.17.3: Question 4. Use the formula in the Key Idea box to calculate P(BIA). Verify that the answer agrees with what you found earlier. Submit ACTIVTIPTON 5.11.4: Question 5. Determine the conditions at the us for scented by both colleges ( Hint : You want to calculate P((A and B) I (A or B) ). Use the definition of conditional probability. applied to these compound events.) Submit Independence Now suppose that a friend has applied to colleges C and D. wit espectively, and a 0.40 probability of acceptance to both. es of 0.80 and 0.50. Question 6 Produce a 2 x 2 table of counts for a hypothetical po probabilities. ulation of 1,000 applicants who had these exact Definition Two events A and B are said to be independent if P( B | A) = P(B). In other words, knowing that event A has occurred does not change the probability that event B will occur. If two events are not independent, they are said to be dependent in other words, knowing that event A has occurred does not A and not B are also independent as are independent, as are A and not B, and so on PARTICIPATION 5.17.5: 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? Butimit ACTIVITY learning about acceptance by one college change the probability of acceptance to the other? ANTICIPATION | 5.17.7: Question 9. Is acceptance by college Cinder tance by college D? Justify your answer RETTOCUPATION | 5.11.8 Question 10. Is acceptance by college A indep Submit Multiplication Rule Now suppose that you have applied to colleges G and had to colleges G and H. You have a 0.70 probability of acceptance by G. and th conditional probability of acceptance by H given acceptance by G is 0.90. The general multiplication rule is four simply by rearranging the equation for condit Definition The general multiplication rule allows you to calculate the probability that two events both occur the first P( An B ) = P(A) x P(B | 4) = P(B) * P(A B). y of the other event given Note : If A and B are indep independent events P(An B) = P(A) x P(B). multiplication rule for ACTIVTIPATION | 5.11.9: Question 11. Subenit