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 not independent because the coin came up heads both times. 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 independent because each coin toss cannot have any outcomes in common.

Explanation:

Step1: Define independent events

Two events $A$ and $B$ are independent if the probability of $B$ occurring is the same whether or not $A$ has occurred, i.e., $P(B|A)=P(B)$. In the case of coin - tosses, the result of the first coin - toss does not affect the result of the second coin - toss.

Step2: Analyze coin - toss situation

When we toss a fair coin, the probability of getting a head on any single toss is $\frac{1}{2}$. The outcome of the second coin - toss is not influenced by the outcome of the first coin - toss. For example, if the first coin shows a head, the probability that the second coin shows a head is still $\frac{1}{2}$, and if the first coin shows a tail, the probability that the second coin shows a head is also $\frac{1}{2}$.

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.