For the given pair of events A and B, complete parts (a) and (b) below. A: When
ID: 3222140 • Letter: F
Question
For the given pair of events A and B, complete parts (a) and (b) below. A: When a page is randomly selected and ripped from a 10-page document and destroyed, it is page 9. B: When a different page is randomly selected and ripped from the document, it is page 7. a. Determine whether events A and B are independent or dependent. (If two events are technically dependent but can be treated as if they are independent according to the 5% guideline, consider them to be independent.) b. Find P(A and B), the probability that events A and B both occur.
Explanation / Answer
In the given problem, the event A and B is defined as
A: When a page is randomly selected and ripped from a 10-page document and destroyed, it is page 9.
B: When a different page is randomly selected and ripped from the document, it is page 7.
(a)
The two events are dependent as once the page 9 is selected and destoyed it cannot be replaced in the 2nd event B i.e. the occurence of the 1st event affects the occurence of the 2nd event. Thus this occurs without replacement and hence the two events A and B are not independent.
Therefore A and B are dependent.
Probability of occurence of A is, P(A) = 1/10 and
Probability of occurence of B is, P(B|A) = 1/9 (as one page is already destroyed)
(Since both the events A and B are dependent, we are interested in finding the probability of the event B given that the event A has already occured. During the event A, the book had 10 pages and thus selecting a page from the 10 pages have equal probability of 1/10. But after event A has occured, the no of pages of the book is now only 9 as the destroyed page cannot be replaced. Thus the probability of now selecting a page from the 9 pages is equal to 1/9.)
(b)
Probability that the events A and B both occur is given by,
P(A and B) = P(A)*P(B given that A has already occured)
= P(A)*P(B|A)
= (1/10)*(1/9)
= 1/90
= 0.0111