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)=0.61
,  P(D)=0.30
and  P(D|B)=0.10
.Find  P(D∣∣B′)
.

Given the problem, we need to find the probability \( P(D \mid B') \), where:

- \( P(B) = 0.61 \) is the probability that the student checks out a book.
- \( P(D) = 0.30 \) is the probability that the student checks out a DVD.
- \( P(D \mid B) = 0.10 \) is the probability that the student checks out a DVD given that they checked out a book.
- \( B' \) denotes the event that the student does **not** check out a book.

To find \( P(D \mid B') \), we can use the law of total probability and Bayes' theorem.

### Step 1: Law of Total Probability
We know:
\[
P(D) = P(D \mid B)P(B) + P(D \mid B')P(B')
\]
Where \( P(B') = 1 - P(B) = 1 - 0.61 = 0.39 \).

Substituting the known values:
\[
0.30 = (0.10)(0.61) + P(D \mid B')(0.39)
\]
\[
0.30 = 0.061 + 0.39 \times P(D \mid B')
\]

### Step 2: Solve for \( P(D \mid B') \)
Rearrange the equation to solve for \( P(D \mid B') \):
\[
0.39 \times P(D \mid B') = 0.30 - 0.061
\]
\[
0.39 \times P(D \mid B') = 0.239
\]
\[
P(D \mid B') = \frac{0.239}{0.39} \approx 0.6128
\]

So, the probability that the student checks out a DVD given that they did not check out a book, \( P(D \mid B') \), is approximately \( 0.6128 \).

Would you like more details on this solution, or do you have any further questions?

Here are 8 related questions you might consider:
1. What is the probability that the student checks out a book or a DVD?
2. How would you calculate \( P(B \cap D') \)?
3. What is the meaning of the law of total probability?
4. Can you explain how Bayes' theorem is applied in conditional probability problems?
5. How would \( P(D \mid B') \) change if \( P(D \mid B) \) were 0.2?
6. What are independent events in probability?
7. How do you calculate \( P(D \cap B') \)?
8. Can you calculate \( P(B \mid D) \) given \( P(D) = 0.30 \) and \( P(D \mid B) = 0.10 \)?

**Tip:** In probability, understanding the difference between mutually exclusive and independent events is crucial for correctly applying formulas.

Learn how to calculate conditional probability involving events B and D in a student library scenario. Understand the probabilities of checking out a book and a DVD, and how to apply the law of total probability and Bayes' theorem. Suitable for advanced high school students and anyone interested in probability theory.

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