5.98 A lottery payoff. A $1 bet in a state lottery’s Pick 3 game pays $500 if the three-digit number you choose exactly matches the winning number, which is drawn at random. Here is the distribution of the payoff X:

340

Payoff X $0 $500
Probability 0.999 0.001

Each day’s drawing is independent of other drawings.

  1. (a) Joe buys a Pick 3 ticket twice a week. The number of times he wins follows a B(104, 0.001) distribution. Using the Poisson approximation to the binomial, what is the probability that he wins at least once?

  2. (b) The exact binomial probability is 0.0988. How accurate is the Poisson approximation here?

  3. (c) If Joe pays $5 a ticket, he needs to win at least twice a year to come out ahead. Using the Poisson approximation, what is the probability that Joe comes out ahead?