Answered

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5. In your backpack, there are three math books, two science books, and five notebooks. If you reach into your backpack and select a math book, and then reach in and select a notebook, are these independent events? Explain.

A. Yes. The probability of drawing a notebook was the same regardless of replacing the math book.

B. No. These are dependent events, and you can calculate the probability of this outcome using the formula [tex]P(A \, \text{and} \, B) = P(A) \cdot P(B|A)[/tex].

C. No. You did not replace the first book, so that changed the probability that you would draw a notebook second.

D. Yes. You can use the formula [tex]P(A \, \text{or} \, B) = P(A) + P(B) - P(A \, \text{and} \, B)[/tex] to calculate the probability of this outcome.

Sagot :

GinnyAnswer
Sure, let's break this down step-by-step.

1. Calculate the total number of items in the backpack:
- Math books: 3
- Science books: 2
- Notebooks: 5
- Total items = 3 + 2 + 5 = 10

2. Calculate the probability of selecting a math book first:
- There are 3 math books out of a total of 10 items.
- Probability of selecting a math book = 3/10 = 0.3

3. After selecting a math book, the total number of items in the backpack decreases by 1:
- New total items = 10 - 1 = 9

4. Calculate the probability of selecting a notebook after a math book has been selected:
- There are still 5 notebooks, but now out of 9 remaining items.
- Probability of selecting a notebook = 5/9 ≈ 0.5556

5. Are these events independent?
- To determine if selecting a math book and then a notebook are independent events, we need to check if the probability of selecting a notebook is the same regardless of whether the math book was selected first or not.
- Initially, the probability of selecting a notebook = 5/10 = 0.5
- After one math book is removed, the probability changes to 5/9 ≈ 0.5556
- Since 0.5556 ≠ 0.5, the probability of selecting a notebook has changed after the math book was removed. Therefore, these events are dependent.

Conclusion:
No, these are dependent events. The probability of drawing a notebook is affected by whether or not a math book was selected first. The correct answer is:

No. You did not replace the first book, so that changed the probability that you would draw a notebook second.
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