- What are three ways to determine probabilities?
Ans: Theoretical, empirical, and subjective.
- What is wrong with this probability distribution?
| Outcome | Probability |
| 10 | 0.4 |
| 20 | 1.2 |
| 30 | -0.3 |
| 90 | 0.0 |
Ans: Probabilities must be between 0 and 1, inclusive, and all
of the probabilities must add up to 1.
- What is a random variable?
Ans: The process of choosing a random number.
- Consider this random variable: flip a coin three times and count the
number of heads. If you get 3 heads, you win 10$. If you get 2 heads,
you win 1$. If you get 1 head, you lose 3$. If you get 0 heads, you
lose 7$. What is the probability distribution?
Ans:
| NHeads | Payoff | Probability |
| 0 | -7 | 1/8 |
| 1 | -3 | 3/8 |
| 2 | +1 | 3/8 |
| 3 | +10 | 1/8 |
- What is another term for risk?
Ans: Expected value. (Positive risk is good, negative risk is bad.)
- What is the risk of the random variable in Problem 4?
Ans: risk = (-7)(1/8) + (-3)(3/8) + 1(3/8) + 10(1/8) = -3/8 = -0.375.
On the average, you will lose 37.5¢ each time you play the game.
- The probability of being struck by lightning in a given year
is 1 / 750,000. How many persons in a million, on the average,
will be struck by lightning at least once in 60 years?
Ans: 1 - (1 - p)n = 1 - (1 - 1/750,000)60
= 7.99969 × 10-5. Multiplying this by 1 million
(106) gives 79.9969 = 80.