6.2 Exercise 11 — Applied Probability

On a Tuesday evening, the Houston Rockets, the Orlando Magic, and the Chicago Bulls all have games (but not with one another). Let A be the event the Rockets win, B be the event the Magic win, and C be the event the Bulls win. Suppose the class $\left\{A , B , C\right\}$ is independent, with respective probabilities 0.75, 0.70 0.8. Ellen's boyfriend is a rabid Rockets fan, who does not like the Magic. He wants to bet on the games. She decides to take him up on his bets as follows:

$10 to 5 on the Rockets --- i.e. She loses five if the Rockets win and gains ten if they lose
$10 to 5 against the Magic
even $5 to 5 on the Bulls.

Ellen's winning may be expressed as the random variable

X = - 5 I_{A} + 10 I_{A^{c}} + 10 I_{B} - 5 I_{B^{c}} - 5 I_{C} + 5 I_{C^{c}} = - 15 I_{A} + 15 I_{B} - 10 I_{C} + 10

Determine the distribution for X. What are the probabilities Ellen loses money, breaks even, or comes out ahead?

Pfeiffer, Applied Probability, 6.2 Problems on Random Variables and Probabilities, exercise 11

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