A coin is tossed three times and the results are recorded. What is the probability of getting:(a) Three heads.(b) At least two heads.(c) At most one tail

(a) Three heads:

The probability of getting a head (H) or a tail (T) in a single toss of a fair coin is 1/2.

Since the tosses are independent, the probability of getting three heads is the product of the probabilities of getting a head in each toss.

So, P(HHH) = P(H) * P(H) * P(H) = (1/2) * (1/2) * (1/2) = 1/8.

(b) At least two heads:

This means we can have either two heads and one tail or three heads.

The number of ways to get two heads and one tail in three tosses is 3 (HTH, HHT, THH).

So, P(at least two heads) = P(HHT or HTH or THH or HHH) = [3 * P(HHT) + P(HHH)] = [3 * (1/2) * (1/2) * (1/2) + (1/2) * (1/2) * (1/2)] = [3/8 + 1/8] = 4/8 = 1/2.

(c) At most one tail:

This means we can have either no tails (three heads) or one tail (two heads and one tail).

So, P(at most one tail) = P(HHH or HHT or HTH or THH) = [P(HHH) + 3 * P(HHT)] = [(1/2) * (1/2) * (1/2) + 3 * (1/2) * (1/2) * (1/2)] = [1/8 + 3/8] = 4/8 = 1/2.