Question

A fair coin is tossed three times. A player wins $1 if the first toss is a head, but loses $1 if the first toss is a tail. Similarly, the player wins $2 if the second toss is a head, but loses $2 if the second toss is a tail, and wins or loses $3 according to the result of the third toss. Let the random variable X be the total winnings after the three tosses (possibly a negative value if losses are incurred). (a) Construct the probability mass function

Answer 1

There are 8 possible outcomes with corresponding winnings:

HHH ==> 1 + 2 + 3 = 6

THH ==> -1 + 2 + 3 = 4

HTH ==> 1 - 2 + 3 = 2

HHT ==> 1 + 2 - 3 = 0

TTH ==> -1 - 2 + 3 = 0

THT ==> -1 + 2 - 3 = -2

HTT ==> 1 - 2 - 3 = -4

TTT ==> -1 - 2 - 3 = -6

Then

`P(X=x)=\begin{cases}\frac18&\text{for }x\in\{-6,-4,-2,2,4,6\}\\\frac14&\text{for }x=0\\0&\text{otherwise}\end{cases}`