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Kaitlin is playing a game in which she spins a spinner with 6 equal-sized slices numbered 1 through 6. The spinner stops on a numbered slice at random.This game is this: Kaitlin spins the spinner once. She wins $1 if the spinner stops on the number 1, $4 if the spinner stops on the number 2, $7 if the spinner stops on the number 3, and $10 if the spinner stops on the number 4. She loses $11 if the spinner stops on 5 or 6.(If necessary, consult a list of formulas.)(a) Find the expected value of playing the game.dollars(b) What can Kaitlin expect in the long run, after playing the game many times?Kaitlin can expect to gain money.Shecanexpecttowindollarsperspin.Kaitlin can expect to lose money.Shecanexpecttolosedollarsperspin.Kaitlin can expect to break even (neither gain nor lose money).

Question

Kaitlin is playing a game in which she spins a spinner with 6 equal-sized slices numbered 1 through 6. The spinner stops on a numbered slice at random.This game is this: Kaitlin spins the spinner once. She wins 1ifthespinnerstopsonthenumber1,1 if the spinner stops on the number 1, 4 if the spinner stops on the number 2, 7ifthespinnerstopsonthenumber3,and7 if the spinner stops on the number 3, and 10 if the spinner stops on the number 4. She loses $11 if the spinner stops on 5 or 6.(If necessary, consult a list of formulas.)(a) Find the expected value of playing the game.dollars(b) What can Kaitlin expect in the long run, after playing the game many times?Kaitlin can expect to gain money.Shecanexpecttowindollarsperspin.Kaitlin can expect to lose money.Shecanexpecttolosedollarsperspin.Kaitlin can expect to break even (neither gain nor lose money).

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Solution 1

To find the expected value of playing the game, we need to multiply each outcome by its probability and then sum these values.

The spinner has 6 equal-sized slices, so the probability of landing on any given number is 1/6.

(a) The expected value is calculated as follows:

(1/6 * 1)+(1/6∗1) + (1/6 * 4) + (1/6 * 7)+(1/6∗7) + (1/6 * 10) - (1/6 * 11)−(1/6∗11) - (1/6 * 11) = $0.50

So, the expected value of playing the game is $0.50.

(b) In the long run, after playing the game many times, Kaitlin can expect to gain $0.50 per spin. This is because the expected value is a long-term average, and it represents the average amount she would win (or lose) per game if she played many games.

This problem has been solved

Solution 2

To find the expected value of playing the game, we need to multiply each outcome by its probability and then sum these values.

The probability of landing on any given number is 1/6 because there are 6 equally likely outcomes.

(a) Expected value calculation:

  • If the spinner stops on 1, Kaitlin wins 1.Theexpectedvaluefromthisis1∗(1/6)=1. The expected value from this is 1*(1/6) = 0.17.
  • If the spinner stops on 2, Kaitlin wins 4.Theexpectedvaluefromthisis4∗(1/6)=4. The expected value from this is 4*(1/6) = 0.67.
  • If the spinner stops on 3, Kaitlin wins 7.Theexpectedvaluefromthisis7∗(1/6)=7. The expected value from this is 7*(1/6) = 1.17.

This problem has been solved

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