Question

Thuy rolls a number cube 7 times. Which expression represents the probability of rolling a 4 exactly 2 times? P (k successes) = Subscript n Baseline C Subscript k Baseline p Superscript k Baseline (1 minus p) Superscript n minus k. Subscript n Baseline C Subscript k Baseline = StartFraction n factorial Over (n minus k) factorial times k factorial EndFraction Subscript 7 Baseline C Subscript 5 Baseline (one-sixth) squared (one-sixth) Superscript 5 Subscript 7 Baseline C Subscript 5 Baseline (one-sixth) Superscript 5 Baseline (five-sixths) squared Subscript 7 Baseline C Subscript 2 Baseline (one-sixth) squared (five-eighths) Superscript 5 Subscript 7 Baseline C Subscript 2 Baseline (two-sixths) squared (four-sixths) Superscript 5

Answer 1

The probability of rolling a 4 exactly two times out of seven rolls of a number cube is calculated using the binomial probability formula, with 2 successes and a probability of 1/6 per trial. The correct expression is 7C2 × (1/6)² × (5/6)⁵.

Step-by-step explanation:

The probability of rolling a 4 exactly 2 times when rolling a number cube 7 times can be calculated using the binomial probability formula: P(k successes) = nCk × p^k × (1-p)^(n-k), where n is the total number of trials, k is the number of desired successes, p is the probability of success on a single trial, and nCk is the number of combinations of n items taken k at a time.

To find this, we first need to identify the values of p and k. Since a six-sided die has an equal chance of landing on any number, the probability of rolling a 4 (or any specific number) on a single roll is 1/6. For exactly 2 successes (rolling a 4), k=2 and n=7 because the die is rolled 7 times. Therefore, the correct expression using the binomial probability formula is:

7C2 × (1/6)^2 × (5/6)^(7-2)

This represents the number of ways to choose 2 successes out of 7 trials, multiplied by the probability of success raised to the number of successes, and the probability of failure raised to the number of failures.