g.pr.10.6 (hc)

you want to play a game that is \$5 to play. you will roll a fair die and spin a spinner with 4 colors: red, blue, yellow, and green. there are 2 ways to win:

* roll an even number and spin red - win \$3
* roll a 3 and spin yellow or green - win \$15

find the expected value of the game.

Question

g.pr.10.6 (hc)

you want to play a game that is \$5 to play. you will roll a fair die and spin a spinner with 4 colors: red, blue, yellow, and green. there are 2 ways to win:

* roll an even number and spin red - win \$3
* roll a 3 and spin yellow or green - win \$15

find the expected value of the game.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Expected Value", "Probability of Compound Events", "Net Gain" ], "new_concepts": [], "current_concepts": [ "Expected Value", "Probability of Compound Events", "Net Gain" ] } ## Calculate probabilities of winning outcomes $$ \begin{aligned} P(\text{Even and Red}) &= P(\text{Even}) \times P(\text{Red}) = \frac{3}{6} \times \frac{1}{4} = \frac{3}{24} = \frac{1}{8} \ P(\text{3 and (Yellow or Green)}) &= P(3) \times P(\text{Yellow or Green}) = \frac{1}{6} \times \frac{2}{4} = \frac{2}{24} = \frac{1}{12} \end{aligned} $$ ## Calculate probability of losing $$ \begin{aligned} P(\text{Lose}) &= 1 - P(\text{Even and Red}) - P(\text{3 and (Yellow or Green)}) \ &= 1 - \frac{3}{24} - \frac{2}{24} = \frac{19}{24} \end{aligned} $$ ## Calculate expected value of net gain $$ \begin{aligned} E(X) &= (3 - 5) \times \frac{3}{24} + (15 - 5) \times \frac{2}{24} + (-5) \times \frac{19}{24} \ &= -2 \times \frac{3}{24} + 10 \times \frac{2}{24} - 5 \times \frac{19}{24} \ &= \frac{-6 + 20 - 95}{24} = \frac{-81}{24} = -3.375 \end{aligned} $$ $-\$3.38$ (or a loss of $\$3.38$) { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Statistics", "Expected Value" ] }

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Explanation:

Calculate probabilities of winning outcomes

Calculate probability of losing

Calculate expected value of net gain

Answer: