Help needed thank you very much!

Question

There is a wheel of prizes at the county fair that costs 1 dollar to spin. Use this wheel of prizes to complete this activity.

Each black section takes up 1/4 of the wheel. If you land on that, you walk away with nothing.

The blue section takes up 1/8 of the wheel. If you land on that, you get a small prize (a piece of candy or bookmark).

The purple section takes up 1/8 of the wheel. If you land on that, you get a medium prize (a bracelet or a squirt gun).

The green section takes up 1/12 of the wheel. If you land on that, you get to spin again.

The red section takes up 1/12 of the wheel. If you land on that, you get a large prize (a stuffed animal or t-shirt).

The yellow section takes up 1/30 of the wheel. If you land on that, you get $5.

The orange section takes up 1/30 of the wheel. If you land on that, you get $10.

The white section takes up 1/60 of the wheel. If you land on that, you get the $50 jackpot.

1. Analyze the wheel. Why do you think it was designed the way it was?

2. If a person spins the wheel once, what is the probability that the results will end favorably (they do not land on 0)?

3. What is the probability that a player will land on "small prize" or "medium prize" with one spin?

4. What is the probability that a player will win money with one spin?

5. Joey decides to spin the wheel 60 times. What is the expected outcome of spins? (How many times would it land on each space of the pie?)

6. Use your information from question 5 to determine whether or not this wheel is profitable for the person who owns the booth.

7. What could the booth owner do to make the wheel more profitable for himself?

Answer 1

1)     This is a bit of an open ended question. I think the main thing you want to consider is that the person who operates the booth doesn’t want to lose money, so they’d probably make the ‘no prize’ sections the biggest and the sections with the best prizes (like Jackpot) the smallest.

2)     So let’s list all of the ‘favorable’ results: You can win a small/medium/large prize, money (5$ or 10$), or hit the jack pot. I don’t know if ‘respin’ counts as favorable or not, so we’ll consider both scenarios. Once you know all the ‘favorable’ results, just add all the probabilities for these favorable results together:

a.      The probability of a favorable result = probability of a small prize (1/8) + probability of a medium prize (1/8) + probability of a larger prize (1/12) + probability of winning 5$ (1/30) + probability of winning 10$ (1/30) + the probability of winning the jackpot (1/60). Add these fractions together finding the least common dominator!

b.      The LCD in this case is 120. 1/8 = 15/120. 1/12 = 10/120. 1/30 = 4/120. 1/60 = 2/120.

c.      So we add up 15/120 (Small prize) + 15/120 (Medium Prize) + 10/120 (Large prize) + 4/120 (5$) + 4/120 (10$) + 2/120 (Jackpot) = 50/120 = 5/12

d.      If we add in ‘respin’ (1/12), we add 5/12 + 1/12 = 6/12 = ½

3)     Similar to question 2, but this time we only add the probabilities of landing on a small prize (1/8) and on a medium prize (1/8). 1/8 + 1/8 = 2/8 = ¼

4)     Similar to question 2, but this time we only add the probabilities that have monetary prizes: 5$ (1/30), 10$ (1/30), and the 50$ Jackpot (1/60). Adding these together, we get 1/30 + 1/30 + 1/60 = 5/60 = 1/12

5)     For expected outcomes, just multiply the number of spins by the probability for each outcome. For example, Joey has a 1/30 chance of winning the 5$ prize. So in 60 spins, Joey expects to win this prize 60 * 1/30 = 2 times. You can repeat this for every other space of the pie.

6)     So this question asks you to weigh costs and potential outcomes. You know that 60 spins costs 60$. After, that you add up all the money/prizes you expect to win if you spin 60 times to see if the winnings are greater than the costs. This question is a little confounded by the fact that we don’t know the monetary cost of the prizes—however, if the amount of money one expects to win is higher than the cost, then we already know that this wheel isn’t profitable for the owner.

7)     There isn’t one right answer, but the main goal for the owner is to maximize profit. How could the owner do so? Either by increasing the cost required to spin the wheel or by reducing the expected winnings from spinning the wheel (which can happen in a number of ways—decreasing prizes, increases the ‘no prize slots’, etc.)