The probability that a student checks out both a book and a DVD is:
Pr(D ∩ B) = 0.225
The probability that a student checks out at least one of either a book or a DVD is: P(B OR D) = 0.725
How to find the conditional probability?
Let B be the event that the student checks out a book, and let D be the event that the student checks out a DVD.
P(B) represents the probability of the event that the student checks out a book.
P(D) represents the probability of the event that the student checks out a DVD.
P(D|B) represents the probability of the event that the student checks out a DVD given that they have checked out a book.
P(B AND D) represents the probability that the student checks out both a book and a DVD.
P(B OR D) represents the probability that the student checks out at least one of either a book or a DVD.
We use the following formulas to find the probabilities:
Pr(A ∪ B) = Pr(A) + Pr(B) - Pr(A ∩ B)
Pr(B|A) = Pr(A ∩ B) / Pr(A)
a. To find P(B AND D), we will use the formula:
Pr(B|D) = Pr(D ∩ B) / Pr(D), which can be rearranged as:
Pr(D ∩ B) = Pr(B|D) * Pr(D).
We are given that:
Pr(D|B) = 0.5
Pr(B) = 0.50
Pr(D) = 0.45
Therefore:
Pr(D ∩ B) = Pr(B|D) * Pr(D)
= 0.50 * 0.45
= 0.225
Hence, P(B AND D) = 0.225
b. To find P(B OR D), we will use the formula:
Pr(A ∪ B) = Pr(A) + Pr(B) - Pr(A ∩ B).
We are given that:
Pr(B) = 0.50,
Pr(D) = 0.45
Pr(D ∩ B) = 0.225
Therefore,
P(B OR D) = P(B U D) = Pr(B) + Pr(D) - P(D ∩ B)
= 0.50 + 0.45 - 0.225
= 0.725
Hence, P(B OR D) = 0.725
Therefore, the probability that a student checks out both a book and a DVD is 0.225, and the probability that a student checks out at least one of either a book or a DVD is 0.725
Complete question is:
A student goes to the library. Let events B = the student checks out a book and D = the student check out a DVD. Suppose that P(B) = 0.50, P(D) = 0.45 and P(D|B) = 0.50.
a. Find P(B AND D).
b. Find P(B OR D).