Question

A professional bowler is practicing for a tournament. On any given bowl, he has a 60 percent probability of getting a strike.

a) At the beginning of his practice, he likes to bowl until he fails to make a strike. How many times is he expected to bowl until he gets his first non-strike?
b) A perfect game consists of bowling 12 straight strikes. How likely is this bowler to bowl a perfect game?
c) In a game, which is 12 bowls, how likely is the bowler to get 9 or more strikes?
d) The bowler has pledged to give a certain amount to charity for every strike he gets during the tournament. He expects to bowl a total of 108 times over the course of the tournament. How much money should he give per strike if he wants to end up giving $1000 to charity?

Answer 1

For the charity donation, the bowler should donate approximately $15.43 per strike to meet the $1000 total.

Step-by-step explanation:

The professional bowler's game can be represented mathematically using probability concepts. Here are the solutions to the student's questions:

1. The expected number of times the bowler bowls until he gets a non-strike can be found using the formula for the expectancy E(X) of a geometric distribution, which is 1/p. Here, p is the probability of a strike, which is 0.60. Thus, E(X) = 1/0.60 = 1.67 bowls. Since a bowler can't bowl a fraction of a time, we expect he bowls 2 times before getting a non-strike.
2. The probability of bowling a perfect game, which consists of 12 strikes in a row, is (0.60)^12, as each bowl is independent of the others. So, the likelihood of a perfect game is 0.60^12 = 0.00217678, or about 0.218%.
3. To calculate the probability of the bowler getting 9 or more strikes in a game of 12 bowls, we use the binomial distribution. The sum of the probabilities of getting exactly 9, 10, 11, and 12 strikes gives us the total probability for this scenario.
4. If the bowler wants to donate a total of $1000 to charity over the course of bowling 108 times and expects to strike 60% of the time, he would be expected to strike 0.60 * 108 = 64.8 times. The amount per strike to donate would be $1000 / 64.8 ≈ $15.43.