Answer These Questions

Question

LINK TO WHEEL https://www.mdlp.org/members/courses/resources/practicals/ma2010/images/402u4p1_1.gif

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

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

3. What is the probability that a player will land on "small prize" or "medium prize" with one spin? Show your work and explain your process.

4. What is the probability that a player will win money with one spin? Show your work and explain your process.

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

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

7. What could the booth owner do to make the wheel more profitable for himself? You may include a drawing of the more profitable wheel if you would like. Explain your reasoning in complete sentences.

Answer 1

1. The wheel was designed the way it was to provide a random outcome from every spin. Categories of increasing value are smaller and smaller sections of the the wheel so that it is less likely that they will be landed on. This gives a fair chance of landing on something, but only a very small chance of getting the best prize.
2. Looking at the wheel, it appears to be split into quarters. 2 quarters of the wheel are 0 (2/4) and 2 quarters have prizes on them (2/4). this means that the is a 50% chance that the wheel will land on a non-zero category (2/4 = 1/2 = 50%).
3. The small and medium prize categories share one quarter of the wheel (1/4). This evaluates to 25% of the wheel's area and would therefore have a 25% chance of being landed on.
4. Assuming the 5, 10, and Jackpot categories would each win the player money, the area they represent on the wheel would be the chance that a player would win money. Since the three money categories take up 1/3 of the top quarter (1/4), the fraction the represent would be 1/3 * 1/4 which equals 1/12. 1/12 is equivalent to 8.33% and that means that a player has an 8.33% chance to win money.
5. If Joey spins the wheel 60 times, the amount of times it would land on each space would be the fraction that space takes up multiplied by the number of spins (60). That means the the 0 category (1/2 of the area) would be landed on 60 * 1/2 = 30 times. the medium and small prizes would each be landed on 1/2 * 1/4 * 60 = 7.5 times (one would be 8 and the other 7 since you can't have a half spin). Large prize and re-spin would each be landed on 1/3 * 1/4 * 60 = 5 times. the 5 and 10 prizes would each be landed on 2/5 * 1/3 * 1/4 * 60 = 2 times. Lastly, the jackpot would be landed on 1/5 * 1/3 * 1/4 * 60 = 1 time.
6. This wheel could be either profitable or unprofitable for the person who owns the booth. If the cost to spin the wheel 60 times is less that 1 Jackpot + 2 5 Prizes + 2 10 Prizes + 5 large prizes + 7.5 medium prizes + 7.5 small prizes + whatever was landed on from the 5 extra spins, then the booth is unprofitable. If he charges more than that amount, then the booth is profitable.
7. To make the wheel more profitable, the owner could either charge more for each spin or shrink the prize categories of the wheel. If he made each one half the size, the people would win prizes half as much.