Question

3. Let T be a random variable giving the number of heads plus the number of tails in three tosses of a coin. List the elements of the sample space S for the three tosses of the coin and assign a value to each sample point.

Answer 1

The sample space S for the three tosses of the coin is:

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

For 3 tosses, possibilities are:

HHH: 3 heads, 0 tails

t = (heads + tails = 3 + 0 = 3)

HHT: 2 heads, 1 tail

t = (heads + tails = 2 + 1 = 3)

HTH: 2 heads, 1 tail

t = (heads + tails = 2 + 1 = 3)

HTT: 1 head, 2 tails

t = (heads + tails = 1 + 2 = 3)

THH: 2 heads, 1 tail

t = (heads + tails = 2 + 1 = 3)

THT: 1 head + 2 tails

t = (heads + tails = 1 + 2 = 3)

TTH: 1 head, 2 tails

t = (heads + tails = 1 + 2 = 3)

TTT: 0 heads, 3 tails

t = (heads + tails = 0 + 3 = 3)

$P(t=3) = P(HHH) + P(HHT) + P(HTH) + P(HTT) + P(THH) + P(THT) + P(TTH) + P(TTT) \\ = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \\ = 8 \times \frac{1}{8} \\ = 1$