g.pr.10.6 (hc)

a game is played where you draw a card from a standard deck and spin a wheel with the numbers 1-10. it costs \$5 to play. the winning plays are below:

* draw a diamond and spin a 2 or 4 - win \$5
* draw a jack and spin an odd number - win \$8
* draw a red 7 and spin a 10 - win \$30

determine the expected value for the game.

Question

g.pr.10.6 (hc)

a game is played where you draw a card from a standard deck and spin a wheel with the numbers 1-10. it costs \$5 to play. the winning plays are below:

* draw a diamond and spin a 2 or 4 - win \$5
* draw a jack and spin an odd number - win \$8
* draw a red 7 and spin a 10 - win \$30

determine the expected value for the game.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Independent Events", "Expected Value", "Net Gain" ], "new_concepts": [], "current_concepts": [ "Independent Events", "Expected Value", "Net Gain" ] } ## Calculate probabilities of winning events $$ \begin{aligned} P(\text{Diamond and } \{2, 4\}) &= \frac{13}{52} \times \frac{2}{10} = \frac{1}{4} \times \frac{1}{5} = \frac{1}{20} = 0.05 \ P(\text{Jack and odd}) &= \frac{4}{52} \times \frac{5}{10} = \frac{1}{13} \times \frac{1}{2} = \frac{1}{26} \approx 0.03846 \ P(\text{Red 7 and } 10) &= \frac{2}{52} \times \frac{1}{10} = \frac{1}{26} \times \frac{1}{10} = \frac{1}{260} \approx 0.00385 \end{aligned} $$ ## Calculate expected winnings $$ \begin{aligned} E(\text{Winnings}) &= 5 \times \frac{1}{20} + 8 \times \frac{1}{26} + 30 \times \frac{1}{260} \ &= \frac{1}{4} + \frac{4}{13} + \frac{3}{26} \ &= \frac{1}{4} + \frac{8}{26} + \frac{3}{26} \ &= \frac{1}{4} + \frac{11}{26} \ &= \frac{13}{52} + \frac{22}{52} = \frac{35}{52} \approx 0.6731 \end{aligned} $$ ## Calculate expected value of the game $$ \begin{aligned} E(\text{Game}) &= E(\text{Winnings}) - \text{Cost} \ &= \frac{35}{52} - 5 \ &= \frac{35 - 260}{52} \ &= -\frac{225}{52} \approx -4.33 \end{aligned} $$ $-\$4.33$ (or $-\frac{225}{52}$ dollars) { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Statistics", "Expected Value" ] }

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Explanation:

Calculate probabilities of winning events

Calculate expected winnings

Calculate expected value of the game

Answer: