Question

a coin will be tossed three times, and each toss will be recorded as heads (h) or tails (t).
give the sample space describing all possible outcomes.
then give all of the outcomes for the event that the second toss is heads.
use the format hth to mean that the first toss is heads, the second is tails, and the third is heads.
if there is more than one element in the set, separate them with commas.
sample space:
event that the second toss is heads:

Explanation:

Step1: Find the sample space

Each toss has 2 possibilities. For 3 - tosses, by the multiplication principle, the total number of outcomes is \(2\times2\times2 = 8\).
List all combinations:

• When the first toss is \(H\):
• If the second toss is \(H\):
• If the third toss is \(H\): \(HHH\)
• If the third toss is \(T\): \(HHT\)
• If the second toss is \(T\):
• If the third toss is \(H\): \(HTH\)
• If the third toss is \(T\): \(HTT\)
• When the first toss is \(T\):
• If the second toss is \(H\):
• If the third toss is \(H\): \(THH\)
• If the third toss is \(T\): \(THT\)
• If the second toss is \(T\):
• If the third toss is \(H\): \(TTH\)
• If the third toss is \(T\): \(TTT\)

So the sample space \(S=\{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}\)

Step2: Find the event that the second toss is heads

Fix the second - position as \(H\).

• When the first toss is \(H\):
• If the third toss is \(H\): \(HHH\)
• If the third toss is \(T\): \(HHT\)
• When the first toss is \(T\):
• If the third toss is \(H\): \(THH\)
• If the third toss is \(T\): \(THT\)

So the event \(E=\{HHH, HHT, THH, THT\}\)

Answer:

Sample space: \(HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\)
Event that the second toss is heads: \(HHH, HHT, THH, THT\)