Question

g.pr.10.6 (mc)

rachel 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, yellow, and orange evenly spread out. the game costs \\$5 to play and there are 3 ways to win.

• roll an odd number and spin red - win \\$5
• roll a 2 and spin blue or green - win \\$10
• roll a 6 and spin yellow - win \\$20

find the probability of rolling an odd number and spinning red.

\\(\bigcirc\\) \\(\frac{1}{3}\\)
\\(\bigcirc\\) \\(\frac{1}{10}\\)
\\(\bigcirc\\) \\(\frac{7}{10}\\)
\\(\bigcirc\\) \\(\frac{1}{5}\\)

Explanation:

<pre_analysis>
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<reasoning>

Determine individual probabilities of the independent events

\[

$$\begin{aligned} P(\text{odd number}) &= \frac{3}{6} = \frac{1}{2} \\ P(\text{red}) &= \frac{1}{5} \end{aligned}$$

\]

Calculate the joint probability of both events occurring

\[

$$\begin{aligned} P(\text{odd number and red}) &= P(\text{odd number}) \times P(\text{red}) \\ &= \frac{1}{2} \times \frac{1}{5} \\ &= \frac{1}{10} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{1}{3}\)</mcq-option>
<mcq-correct>(B) \(\frac{1}{10}\)</mcq-correct>
<mcq-option>(C) \(\frac{7}{10}\)</mcq-option>
<mcq-option>(D) \(\frac{1}{5}\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Statistics",
"Independent Events"
]
}
</post_analysis>

Answer:

<pre_analysis>
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"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
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"Probability Formulas"
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"new_concepts": [],
"current_concepts": [
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}
</pre_analysis>

<reasoning>

Determine individual probabilities of the independent events

\[

$$\begin{aligned} P(\text{odd number}) &= \frac{3}{6} = \frac{1}{2} \\ P(\text{red}) &= \frac{1}{5} \end{aligned}$$

\]

Calculate the joint probability of both events occurring

\[

$$\begin{aligned} P(\text{odd number and red}) &= P(\text{odd number}) \times P(\text{red}) \\ &= \frac{1}{2} \times \frac{1}{5} \\ &= \frac{1}{10} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{1}{3}\)</mcq-option>
<mcq-correct>(B) \(\frac{1}{10}\)</mcq-correct>
<mcq-option>(C) \(\frac{7}{10}\)</mcq-option>
<mcq-option>(D) \(\frac{1}{5}\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Statistics",
"Independent Events"
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