Question

for the following experiment, determine whether the two given events are independent. a fair coin is tossed twice. the events are \head on the first\ and \head on the second.\ choose the correct answer below. a. the two events are independent because each coin toss cannot have any outcomes in common. b. the two events are not independent because the outcome of the second coin toss depends on the outcome of the first coin toss. c. the two events are independent because the outcome of the second coin toss does not depend on the outcome of the first coin toss. d. the two events are not independent because the coin came up heads both times.

Explanation:

Step1: Recall Independence of Events

Two events \( A \) and \( B \) are independent if \( P(A \cap B) = P(A) \times P(B) \). For a fair coin, the outcome of the first toss (event \( A \): head on first) and the outcome of the second toss (event \( B \): head on second) are independent because the result of one toss does not affect the other. The probability of head on first toss \( P(A)=\frac{1}{2} \), head on second \( P(B)=\frac{1}{2} \), and \( P(A \cap B)=\frac{1}{4} \) (since there are 4 possible outcomes: HH, HT, TH, TT), and \( \frac{1}{2} \times \frac{1}{2}=\frac{1}{4} \), so they are independent. The outcome of the second toss does not depend on the first.

Step2: Analyze Options

• Option A: Wrong, as the events can have common outcomes (HH) and are independent.
• Option B: Wrong, the second toss outcome does not depend on the first.
• Option C: Correct, because the second toss outcome is independent of the first (a fair coin toss is memoryless, so past results don't affect future ones).
• Option D: Wrong, the fact that it came heads both times is a result, not a reason for dependence/independence.

Answer:

C. The two events are independent because the outcome of the second coin toss does not depend on the outcome of the first coin toss.