In Exercises 9 and 10, find the expected net gain to the playe...

Question

In Exercises 9 and 10, find the expected net gain to the player for one play of the game.

It costs $25 to bet on a horse race. The horse has a 1/8 chance of winning and a 1/4 chance of placing second or third. You win $125 if the horse wins and receive your money back if the horse places second or third.

Accepted Answer

Welcome back, everyone. A raffle ticket costs $30. There is a 1 out of 10 chance of winning the grand prize and a 1 out of 5 chance of winning a consolation prize. If you win the grand prize, you'll receive $180. If you win the consolation prize, you get your $30 back. What is the expected net gain for one ticket? A negative $6.50 B-6 dollars, C negative $5.50 and D, negative $3.25. So for this problem, let's recall the expected value formula. E X is equal to the sum of the random variable X values multiplied by their corresponding probabilities. And what we're going to do is simply construct a table. First of all, let's introduce net game. Which corresponds to our random variable X values, and in the second row, we're going to list the corresponding probabilities. So now, what are the possibilities for this problem? Well, first of all, let's recall that the ticket cost C is equal to $30. First of all, we can win the grand prize. Of $180 right, but we had to pay $30. So our first net gain and one is going to be our one of $180 minus our investment of $30 which is $150. And the corresponding probability would be 1 out of 10, right? Then We are going to identify our next possible net gain, which would be $0. It says that if we win the cancellation prize, we get our $30 back, meaning net gain is simply $0. Simply speaking, we invested $30 to buy a ticket and we basically get $30 back. So 30 minus $30 gives us $0 and the probability of this constellation price is 1 out of 5. And of course, we have to consider our final gain of negative. $30. That's in case we lose. How do we identify that probability? Well, since we have a probability distribution table, the probability of not winning. is equal to 1 minus the probability of winning. So basically we have to sum the two probabilities of 1 divided by 10 and 1 divided by 5. This will be the combined probability of winning, and we're subtracting it from 1 to get the probability of losing. So let's find the common denominator, that would be 10. So we have 10 divided by 10 minus 1 divided by 10, minus 2 divided by 10, and this is 7 divided by 10. Well done, and from this table we can identify our expected value. So our expected value is going to be equal to $150 our first x value multiplied by 1 divided by 10, our corresponding probability plus $0 multiplied by 1/5. And Minus, we can say plus minus. Or simply minus 30. Dollars multiplied by 7 divided by 10. This is equal to $-6. Considering the answer choices, we can conclude that B is the correct answer. Thank you for watching.

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Key Concepts

Expected Value

Probability

Net Gain

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