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 2121-page document and destroyed, it is page 1515.
B: When a different page is randomly selected and ripped from the document, it is page 1010.
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.
The probability that events A and B both occur is:
21, 15, 10, 5%
a. To determine if events A and B are independent or dependent, we need to compare the probabilities of each event occurring individually to the probability of both events occurring together.
If events A and B are independent, the probability of both events occurring together (P(A and B)) would be equal to the product of their individual probabilities (P(A) * P(B)).
If events A and B are dependent, the probability of both events occurring together (P(A and B)) would be different from the product of their individual probabilities (P(A) * P(B)).
To find out if events A and B are independent, we need more information about the document and how the pages are ripped.
b. The probability that events A and B both occur (P(A and B)) will be determined by multiplying the probabilities of each event occurring individually, assuming they are independent. However, since we don't have the probabilities for events A and B, we cannot calculate P(A and B) at this time.
To determine whether events A and B are independent or dependent, we need to compare the probability of event A occurring with the probability of event B occurring.
a. Independence: Two events A and B are said to be independent if the occurrence of one event does not affect the probability of the other event occurring. In other words, if the probability of A occurring remains the same whether or not B has already occurred, then events A and B are independent.
To determine independence, we need to find the probability of event A occurring (P(A)) and the probability of event B occurring (P(B)). If the probability of A occurring is not influenced by the occurrence of B and vice versa, we can treat them as independent events.
b. To find the probability that events A and B both occur (P(A and B)), we multiply the probabilities of both events.
P(A and B) = P(A) * P(B)
Let's calculate the probabilities:
Given:
Total number of pages in the document (n) = 2121
Page selected in event A (page A) = 1515
Page selected in event B (page B) = 1010
a. To determine if events A and B are independent or dependent, we need to compare their probabilities.
P(A) = 1/n (Since there is only one page 1515 out of 2121 pages to be selected at random)
P(A) = 1/2121
P(B) = 1/n (Since there is only one page 1010 out of 2121 pages to be selected at random)
P(B) = 1/2121
If P(A) = P(B), then we can treat them as independent events.
b. To find P(A and B), we multiply P(A) and P(B).
P(A and B) = P(A) * P(B)
P(A and B) = (1/2121) * (1/2121)
P(A and B) = 1/4490441
Therefore, the probability that events A and B both occur is 1/4490441.