Consider an experiment of rolling a fair four sided die twice where all the possible outcomes are equally likely. Define the eventsA = 1st roll results in a 1B = Sum of the two rolls is a 7C = 2nd roll results in a 2Which among the following statements are true?Events A and C are independent. Events A, B and C are mutually exclusive.Events A, B and C are exhaustive.P(A | (B ∪ C)) = 1/6
Question
Consider an experiment of rolling a fair four sided die twice where all the possible outcomes are equally likely. Define the eventsA = 1st roll results in a 1B = Sum of the two rolls is a 7C = 2nd roll results in a 2Which among the following statements are true?Events A and C are independent. Events A, B and C are mutually exclusive.Events A, B and C are exhaustive.P(A | (B ∪ C)) = 1/6
Solution
Let's analyze each statement one by one:
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Events A and C are independent: This statement is true. The outcome of the first roll (Event A) does not affect the outcome of the second roll (Event C). They are two separate events.
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Events A, B and C are mutually exclusive: This statement is false. Mutually exclusive means that if one event occurs, the others cannot. However, it's possible for A and C to occur at the same time (rolling a 1 first and a 2 second). Also, B can occur with either A or C (if the first roll is a 1 and the second roll is a 6, or vice versa, the sum is 7).
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Events A, B and C are exhaustive: This statement is also false. Exhaustive events mean that at least one of the events must occur. However, it's possible that none of A, B, or C occur (for example, if the first roll is a 3 and the second roll is a 4).
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P(A | (B ∪ C)) = 1/6: This statement is false. The probability of A given that either B or C has occurred is not 1/6. For B to occur, the sum of the two rolls must be 7, which is impossible with a four-sided die. For C to occur, the second roll must be a 2, which has a 1/4 chance. If either of these occur, the chance of A (the first roll being a 1) is 1/4, not 1/6.
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