1. 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. No. You did not replace the first book, so that changed the probability that you would draw a notebook second.
B. Yes. You can use the formula [tex]P(A[/tex] or [tex]B)=P(A)+P(B)-P(A[/tex] and [tex]B)[/tex] to calculate the probability of this outcome.
C. No. These are dependent events and you can calculate the probability of this outcome using the formula [tex]P(A[/tex] or [tex]B)=P(A) \cdot P(B)[/tex].
D. Yes. The probability of drawing a notebook was the same regardless of replacing the math book.
Answer (2)
Calculate the probability of selecting a math book first: P ( M ) = 10 3 .
Calculate the probability of selecting a notebook second, given a math book was selected first: P ( N ∣ M ) = 9 5 .
Calculate the probability of selecting a notebook second with replacement: P ( N ) = 10 5 = 2 1 .
Since P ( N ∣ M ) = P ( N ) , the events are dependent. The correct answer is: No. You did not replace the first book, so that changed the probability that you would draw a notebook second.
Explanation
Analyze the problem We have a backpack with 3 math books, 2 science books, and 5 notebooks, making a total of 10 items. We want to determine if selecting a math book and then a notebook without replacement are independent events.
Calculate P(M) Let's calculate the probability of selecting a math book first. There are 3 math books out of 10 total items, so the probability is: P ( M ) = 10 3
Calculate P(N|M) Now, let's calculate the probability of selecting a notebook second, given that a math book was selected first and not replaced. After selecting a math book, there are now 9 items left in the backpack, with 5 of them being notebooks. So the conditional probability is: P ( N ∣ M ) = 9 5
Calculate P(N) If the events were independent, the probability of selecting a notebook second would be the same whether or not a math book was selected first. Let's calculate the probability of selecting a notebook if the first item was replaced. In this case, there would still be 10 items in the backpack, with 5 notebooks. So the probability would be: P ( N ) = 10 5 = 2 1
Compare P(N|M) and P(N) Comparing P ( N ∣ M ) and P ( N ) , we have 9 5 and 2 1 . Since 9 5 e q 2 1 , the probability of selecting a notebook changes depending on whether a math book was selected first. Therefore, the events are dependent.
Conclusion The correct explanation is: No. You did not replace the first book, so that changed the probability that you would draw a notebook second.
Examples
This concept of dependent events is crucial in scenarios like drawing cards from a deck. If you draw a card and don't replace it, the probabilities for the next draw change, affecting your chances of getting a specific card. Understanding this helps in making informed decisions in games, statistical analysis, and even in business when assessing sequential probabilities.
The events of selecting a math book and then a notebook are dependent because not replacing the math book changes the probability of drawing a notebook second. Therefore, the correct answer is A. No. You did not replace the first book, so that changed the probability that you would draw a notebook second.
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