Question

a coin is flipped until two heads appear, up to a maximum of three flips. (if two tails are flipped, the coin is still tossed a third time to complete the experiment.) complete parts (a) through (d) below.

b. ( s = {hh, hth, thh, htt, tht, tth, ttt} )
c. ( s = {hhh} )
d. ( s = {hhhh, hhht, hhth, hthh, thhh, hhtt, htht, thht, thth, tthh, htth} )

( n(s) = 7 )
(type a whole number.)

are the outcomes in ( s ) equally likely?

a. yes. the outcomes in ( s ) are equally likely since each outcome occurs with equal probability.
b. no. the outcomes in ( s ) are not equally likely since at least one of the outcomes does not occur with the same probability as all the others.
c. yes. the outcomes in ( s ) are equally likely since the sample space consists of ( n(s) ) simple events.
d. no. the outcomes in ( s ) are not equally likely since the outcomes in the sample space are not mutually exclusive.

Explanation:

To determine if outcomes are equally likely, we analyze their probabilities. For a fair coin, each flip is independent with \( P(H) = P(T) = \frac{1}{2} \).

• \( P(hh) \): The experiment stops at 2 flips (two heads). Probability: \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \).
• \( P(hth) \), \( P(thh) \): Experiment stops at 3 flips (two heads, with first/second tail). Each: \( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \).
• \( P(htt) \), \( P(tht) \), \( P(tth) \), \( P(ttt) \): Experiment continues to 3 flips (fewer than two heads). Each: \( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \).

Outcome \( hh \) has probability \( \frac{1}{4} \), while others have \( \frac{1}{8} \). Thus, not all outcomes have the same probability.

Answer:

B. No. The outcomes in \( S \) are not equally likely since at least one of the outcomes does not occur with the same probability as all the others.