Question

You spin each spinner once. You get 50 if you spin a 2 and a vowel. You get 25 if you spin a 2 and a consonant. You get a 5 if you spin a 1 and a vowel. Everything else earns 0. What is the expected value of this game?

Answer 1

The expected value of this game is 0.64, meaning that you can expect to earn 64 cents per game on average. However, this does not guarantee the same outcome in every game.

Step-by-step explanation:

The expected value of this game can be calculated by multiplying the probability of each outcome by its corresponding value and summing all the results. Let's calculate it step by step:

1. Probability of spinning a 2 and a vowel: There are 5 vowels in the alphabet (a, e, i, o, u) and the probability of spinning a 2 is 1/6. So, the probability of spinning a 2 and a vowel is 1/6 * 5/26 = 5/156.
2. Value of spinning a 2 and a vowel: 50.
3. Probability of spinning a 2 and a consonant: There are 21 consonants in the alphabet and the probability of spinning a 2 is 1/6. So, the probability of spinning a 2 and a consonant is 1/6 * 21/26 = 7/52.
4. Value of spinning a 2 and a consonant: 25.
5. Probability of spinning a 1 and a vowel: There are 5 vowels in the alphabet and the probability of spinning a 1 is 1/6. So, the probability of spinning a 1 and a vowel is 1/6 * 5/26 = 5/156.
6. Value of spinning a 1 and a vowel: 5.
7. Probability of spinning any other outcome: 1 - (5/156 + 7/52) = 1 - 35/156 = 121/156.
8. Value of any other outcome: 0.
9. Expected value = (5/156) * 50 + (7/52) * 25 + (5/156) * 5 + (121/156) * 0 = 0.64.

The expected value of this game is 0.64. This means that, on average, you can expect to earn 64 cents per game if you play it repeatedly. However, it's important to note that the expected value represents the average outcome over the long term and does not guarantee that you will earn exactly 64 cents in every game.