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Chapter 5: Problem 52

When two dice are rolled, is the event "the first die shows a 1 on top" independent of the event "the second die shows a 1 on top"?

Short Answer

Expert verified
Yes, the event 'the first die shows a 1 on top' is independent of the event 'the second die shows a 1 on top'.

Step by step solution

01

Define the Events and Their Probabilities

First, define Event A as 'the first die shows a 1 on top' and Event B as 'the second die shows a 1 on top'. The probability of rolling a 1 on a fair, six-sided die is \(\frac{1}{6}\), so \(P(A) = \frac{1}{6}\) and \(P(B) = \frac{1}{6}\).
02

Calculate the Joint Probability

Next, calculate the joint probability of the two events occurring together, denoted as \(P(A \cap B)\). Since the two dice rolls are independent of each other, the joint probability can be calculated as the product of the individual probabilities, i.e., \(P(A \cap B) = P(A) * P(B) = \frac{1}{6} * \frac{1}{6} = \frac{1}{36}\).
03

Check for Independence

Finally, check if the two events are independent. If the two events are indeed independent, the probability of both events occurring simultaneously is equal to the product of their individual probabilities, i.e., \(P(A \cap B) = P(A) * P(B)\). Since \(\frac{1}{36} = \frac{1}{6} * \frac{1}{6}\), the two events are independent.

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