Answer:
The probability of getting a "six" on one roll of a fair six-sided die is 1/6.
To calculate the probability of getting 5 "sixes" in 7 rolls, we can use the binomial distribution formula, which is:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
where:
P(X = k) is the probability of getting exactly k "successes" in n trials.
(n choose k) is the number of ways to choose k items from a set of n items.
p is the probability of "success" on any one trial.
(1-p) is the probability of "failure" on any one trial.
k is the number of "successes" we are interested in.
n is the total number of trials.
In this case, we want to find the probability of getting 5 "sixes" in 7 rolls, so:
n = 7 (the total number of rolls)
k = 5 (the number of "successes" we want to achieve)
p = 1/6 (the probability of getting a "six" on any one roll)
(1-p) = 5/6 (the probability of not getting a "six" on any one roll)
Using the binomial distribution formula, we can calculate:
P(X = 5) = (7 choose 5) * (1/6)^5 * (5/6)^2
= (7! / (5! * 2!)) * (1/6)^5 * (5/6)^2
= (21 * 1/7776 * 25/36)
= 0.0323
Therefore, the probability of getting exactly 5 "sixes" in 7 rolls of a fair six-sided die is approximately 0.0323, or about 3.23%.