Final answer:
To find the probability of rolling exactly six fours when rolling a die 8 times, we need to use the concept of binomial probability. The probability can be calculated using the formula P(X=k) = nCk * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success. Plugging in the values, we can calculate that the probability is approximately 2.6%.
Step-by-step explanation:
To find the probability of rolling exactly six fours when rolling a die 8 times, we need to use the concept of binomial probability.
A binomial probability can be calculated using the formula:
P(X=k) = nCk * p^k * (1-p)^(n-k)
Where:
- P(X=k) represents the probability of getting exactly k successes
- n is the number of trials (number of times the die is rolled)
- k is the number of successes (number of fours we want to roll)
- p is the probability of getting a success (probability of rolling a four)
In this case, n = 8 (because the die is rolled 8 times), p = 1/6 (because there is one favorable outcome out of 6 possible outcomes), and k = 6 (because we want to roll exactly six fours).
Plugging in these values into the formula, we get:
P(X=6) = 8C6 * (1/6)^6 * (5/6)^(8-6)
To simplify the calculation, we can use the combination formula:
8C6 = 8! / (6!(8-6)!)
Calculating this, we get:
8C6 = 8! / (6! * 2!) = 8 * 7 / 2 = 28
Plugging in this value, along with the probabilities, into the probability formula, we get:
P(X=6) = 28 * (1/6)^6 * (5/6)^2 ≈ 0.026
So, the probability of rolling exactly six fours when rolling a die 8 times is approximately 0.026, or 2.6%.