Questions: Determine whether the following individual events are independent or dependent. Then find the probability of the combined event. Rolling two 4s followed by one 6 on three tosses of a fair die. Choose the correct answer below. (Type an integer or a simplified fraction.) A. The individual events are independent. The probability of the combined event is . B. The individual events are dependent. The probability of the combined event is .

Solution Steps

The events in question are:

1. Rolling a 4 on the first toss.
2. Rolling a 4 on the second toss.
3. Rolling a 6 on the third toss.

Since each roll of a fair die is independent of the others, the outcome of one roll does not affect the outcome of another. Therefore, the individual events are independent.

For a fair die:

• The probability of rolling a 4 on any single toss is \( \frac{1}{6} \).
• The probability of rolling a 6 on any single toss is \( \frac{1}{6} \).

Since the events are independent, the probability of the combined event (rolling two 4s followed by one 6) is the product of the probabilities of each individual event:

\[ P(\text{two 4s followed by one 6}) = P(4) \times P(4) \times P(6) = \frac{1}{6} \times \frac{1}{6} \times \frac{1}{6} = \frac{1}{216}. \]

Final Answer