When rolling a fair six-sided die, the probability of rolling a six is 1/6. Assuming that the dice are fair and independent, we can use the rules of probability to find the probability of rolling certain outcomes when 4 dice are thrown:
i. To find the probability of getting exactly 3 sixes when 4 dice are thrown, we can use the binomial probability formula:
P(exactly k successes in n trials) = (n choose k) * p^k * (1-p)^(n-k)
where n is the number of trials, k is the number of successful outcomes, p is the probability of success on each trial, and (n choose k) is the binomial coefficient.
For this problem, n = 4, k = 3, and p = 1/6.
P(exactly 3 sixes) = (4 choose 3) * (1/6)^3 * (5/6)^1
= 4 * (1/216) * (5/6)
= 5/54
Therefore, the probability of getting exactly 3 sixes when 4 dice are thrown is 5/54.
ii. To find the probability of getting exactly 2 sixes when 4 dice are thrown, we can use the same formula with k = 2:
P(exactly 2 sixes) = (4 choose 2) * (1/6)^2 * (5/6)^2
= 6 * (1/36) * (25/36)
= 25/72
Therefore, the probability of getting exactly 2 sixes when 4 dice are thrown is 25/72.
iii. To find the probability of getting no sixes when 4 dice are thrown, we can use the complement rule:
P(no sixes) = 1 - P(at least one six)
To find the probability of getting at least one six, we can use the complement of getting no sixes:
P(at least one six) = 1 - P(no sixes)
For each die, the probability of not rolling a six is 5/6. Therefore, the probability of getting no sixes on 4 dice is:
P(no sixes) = (5/6)^4
= 625/1296
Therefore, the probability of getting no sixes when 4 dice are thrown is 625/1296, and the probability of getting at least one six is 1 - 625/1296 = 671/1296.