g.pr.10.6 (mc)

florence wants to play a game at a casino where she will roll a fair die and spin a spinner with the colors red, blue, green, and yellow evenly spread out. the game costs \$5 to play and there are 3 ways to win.

* roll a 1 or 2 and spin red or yellow - win \$2
* roll a 3 and spin blue - win \$20
* roll an even number and spin a yellow - win \$5

find the probability of not winning a prize.

Question

g.pr.10.6 (mc)

florence wants to play a game at a casino where she will roll a fair die and spin a spinner with the colors red, blue, green, and yellow evenly spread out. the game costs \$5 to play and there are 3 ways to win.

* roll a 1 or 2 and spin red or yellow - win \$2
* roll a 3 and spin blue - win \$20
* roll an even number and spin a yellow - win \$5

find the probability of not winning a prize.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Probability of Compound Events", "Mutually Exclusive Events" ], "current_concepts": [ "Independent Events", "Probability Formulas", "Complementary Events" ] } ## Determine the total number of outcomes in the sample space $$ \begin{aligned} &\text{Die outcomes: } D = \{1, 2, 3, 4, 5, 6\} \implies |D| = 6\ &\text{Spinner outcomes: } S = \{\text{red}, \text{blue}, \text{green}, \text{yellow}\} \implies |S| = 4\ &\text{Total outcomes: } N = |D| \times |S| = 6 \times 4 = 24 \end{aligned} $$ ## Identify the winning outcomes for each prize condition $$ \begin{aligned} &W_1 = \{1, 2\} \times \{\text{red}, \text{yellow}\} \implies |W_1| = 2 \times 2 = 4\ &W_2 = \{3\} \times \{\text{blue}\} \implies |W_2| = 1 \times 1 = 1\ &W_3 = \{2, 4, 6\} \times \{\text{yellow}\} \implies |W_3| = 3 \times 1 = 3 \end{aligned} $$ ## Calculate the probability of not winning a prize $$ \begin{aligned} &W_1 \cap W_3 = \{(2, \text{yellow})\} \implies |W_1 \cap W_3| = 1\ &W_1 \cap W_2 = \emptyset,\quad W_2 \cap W_3 = \emptyset\ &\text{Total unique winning outcomes: } |W| = |W_1| + |W_2| + |W_3| - |W_1 \cap W_3| = 4 + 1 + 3 - 1 = 7\ &P(\text{not winning}) = 1 - \frac{|W|}{N} = 1 - \frac{7}{24} = \frac{17}{24} \end{aligned} $$ $\frac{17}{24}$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Statistics", "Probability Formulas" ] }

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Question

Explanation:

Determine the total number of outcomes in the sample space

Identify the winning outcomes for each prize condition

Calculate the probability of not winning a prize

Answer: