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.

(c) write the following event in set notation
exactly two heads are tossed.
choose the correct answer below.

a. {hh, hth, thh}

b. {hht, htht, thh}

c. {hh}

d. {hh, hth, thh, htth, thth, tthh}

Explanation:

To determine the set of outcomes with exactly two heads, we analyze the coin - flipping experiment:

1. If the experiment stops when two heads appear:

• The sequence \(hh\) has exactly two heads (and the experiment stops after 2 flips).
• For sequences with 3 flips (because we can have up to 3 flips), we need to have exactly two heads. The possible sequences are \(hth\) (first head, second tail, third head) and \(thh\) (first tail, second head, third head).

1. Now let's analyze the other options:

• Option B: \(\{hht, htht, thh\}\) is incorrect. The sequence \(hht\) has two heads and one tail, but in our experiment, if we get \(hh\) first, the experiment stops, so \(hht\) is not a valid outcome. Also, \(htht\) has more than 3 flips which is not allowed as the maximum number of flips is 3.
• Option C: \(\{hh\}\) is incorrect because it misses the 3 - flip sequences (\(hth\) and \(thh\)) that also have exactly two heads.
• Option D: \(\{hh, hth, thh, htth, thth, tthh\}\) is incorrect. The sequences \(htth\), \(thth\), and \(tthh\) have 4 flips, but our experiment has a maximum of 3 flips, so these are not valid.

Answer:

A. \(\{hh, hth, thh\}\)