a carnival game offers a $100 cash prize for anyone who can break a balloon by throwing a dart at it. It costs $5 to play and you're willing to spend up to $20 trying to win. You estimate that you have about a 10% chance of hitting a balloon on any throw. Create a probability model then find the expected number of darts you'll throw and expected number of winnings.
How do I create a probability model for this question?
A probability model is just a set of all possible outcomes. For this carnival game, I'm assuming that you're not permitted to keep playing after you win, so the two general classes of outcomes are you either win within 4 throws and then stop playing or you don't win within 4 throws. The five specific outcomes are that you can win on your first, second, third, or fourth throw or not win after 4 throws. Each outcome can be assigned a specific probability of taking place, and you can calculate the expected values from these probabilities.
Each individual throw consists of a Bernoulli trial with a probability of success p = 0.1 and a probability of failure q = 1−p = 0.9. For each of the five specific outcomes, you'll need to calculate the probability of achieving that outcome, the number of darts thrown to achieve that outcome, and the winnings for that outcome. For example, in order to win on the third throw, you need to have failed on the first two throws and succeeded on the third:
P(win on the third throw)=q×q×p=q2p=(0.9)2(0.1)=0.081
Darts(win on the third throw)=3
Winnings(win on the third throw)=$100−$5×3=$100-$15=$85
This means that there's an 8.1% chance that you'll throw exactly 3 darts and an 8.1% chance you'll have winnings of exactly $85. Do the same thing for the other four specific outcomes. Don't forget to use "negative winnings" for the case of missing with all four darts. If your five calculated P probabilities add up to one then you've probably found them correctly.
Finally, calculate weighted averages for the number of darts and for your winnings. For each of the five possible outcomes, multiply the values for "Darts" by the probability that the related outcome will occur, and do the same for "Winnings." For the outcome above of winning on the third throw, you would have:
3×0.081 = 0.243 for number of darts
$85×0.081 = $6.885 for winnings
Add these products together for all five outcomes to obtain a weighted average. These weighted averages are your expected values.