Question

a coin is tossed three times. an outcome is represented by a string of the sort htt (meaning a head on the first toss, followed by two tails). the 8 outcomes listed in the table below. note that each outcome has the same probability. for each of the three events in the table, check the outcome(s) that are contained in the event. then, in the last column, enter the probability of the event. event a: a tail on the first toss event b: exactly one head event c: no tails on the first two tosses

Explanation:

Step1: Identify total number of outcomes

The total number of possible outcomes when a coin is tossed 3 - times is \(n = 2\times2\times2=8\) since each toss has 2 possibilities. Each outcome has a probability of \(\frac{1}{8}\).

Step2: Determine outcomes for Event A

Event A is a tail on the first toss. The outcomes are THT, TTH, TTT, THH. The number of outcomes for Event A, \(n_A = 4\). The probability \(P(A)=\frac{n_A}{n}=\frac{4}{8}=\frac{1}{2}\).

Step3: Determine outcomes for Event B

Event B is exactly one head. The outcomes are TTH, THT, HTT. The number of outcomes for Event B, \(n_B = 3\). The probability \(P(B)=\frac{n_B}{n}=\frac{3}{8}\).

Step4: Determine outcomes for Event C

Event C is no tails on the first two tosses. The outcomes are HHH, HHT. The number of outcomes for Event C, \(n_C = 2\). The probability \(P(C)=\frac{n_C}{n}=\frac{2}{8}=\frac{1}{4}\).

Event	Outcomes	Probability
Event B	TTH, THT, HTT	\(\frac{3}{8}\)
Event C	HHH, HHT	\(\frac{1}{4}\)

Answer:

Event A: THT, TTH, TTT, THH; Probability: \(\frac{1}{2}\)
Event B: TTH, THT, HTT; Probability: \(\frac{3}{8}\)
Event C: HHH, HHT; Probability: \(\frac{1}{4}\)