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 $1 to play. the winning plays are below:

* draw a face card and spin a 2-win $5
* draw a queen and spin an odd number-win $10
* draw a red king and spin a 10-win $50

determine the expected value for the game.

( ) -$0.31
( ) $11.25
( ) -0.25
( ) $15

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 $1 to play. the winning plays are below:

* draw a face card and spin a 2-win $5
* draw a queen and spin an odd number-win $10
* draw a red king and spin a 10-win $50

determine the expected value for the game.

( ) -$0.31
( ) $11.25
( ) -0.25
( ) $15

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Expected Value", "Probability Formulas", "Independent Events" ], "new_concepts": [], "current_concepts": [ "Expected Value", "Probability Formulas", "Independent Events", "Net Gain" ] } ## Calculate probabilities of winning events $$ \begin{aligned} &P(\text{Face Card and } 2) = \frac{12}{52} \times \frac{1}{10} = \frac{3}{13} \times \frac{1}{10} = \frac{3}{130} \ &P(\text{Queen and Odd}) = \frac{4}{52} \times \frac{5}{10} = \frac{1}{13} \times \frac{1}{2} = \frac{1}{26} = \frac{5}{130} \ &P(\text{Red King and } 10) = \frac{2}{52} \times \frac{1}{10} = \frac{1}{26} \times \frac{1}{10} = \frac{1}{260} \end{aligned} $$ ## Calculate expected winnings $$ \begin{aligned} E(\text{Winnings}) &= 5 \times P(\text{Face Card and } 2) + 10 \times P(\text{Queen and Odd}) + 50 \times P(\text{Red King and } 10) \ &= 5 \times \frac{3}{130} + 10 \times \frac{5}{130} + 50 \times \frac{1}{260} \ &= \frac{15}{130} + \frac{50}{130} + \frac{25}{130} \ &= \frac{90}{130} = \frac{9}{13} \approx 0.6923 \end{aligned} $$ ## Calculate expected value of the game $$ \begin{aligned} E(\text{Game}) &= E(\text{Winnings}) - \text{Cost to play} \ &= 0.6923 - 1.00 \ &\approx -0.3077 \approx -\$0.31 \end{aligned} $$

(A) -\$0.31

(B) \$11.25

(C) -0.25

(D) \$15

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Statistics", "Expected Value" ] }

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Question

Explanation:

Calculate probabilities of winning events

Calculate expected winnings

Calculate expected value of the game

Answer: