Question

g.pr.10.3 (hc)

a survey is done in a school drama club to see if they want pizza for dinner on long rehearsal days. they were asked if they liked pizza or not. the results are below. what is the probability a female in the club likes pizza? are the events being female and enjoying pizza independent?

results of the survey

	likes pizza	does not like pizza	total
male	16	4	20
total	40	10	50

( ) 48%; yes
( ) 80%; yes
( ) 48%; no
( ) 60%; no

Explanation:

<pre_analysis>
{
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"skills_matched": [
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"concepts_used": [
"Independent Events",
"Probability Formulas",
"Two-Way Frequency Tables"
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"new_concepts": [],
"current_concepts": [
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"Two-Way Frequency Tables",
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</pre_analysis>

<reasoning>

Calculate the conditional probability of a female liking pizza

\[
P(\text{Likes Pizza} \mid \text{Female}) = \frac{\text{Female and Likes Pizza}}{\text{Total Female}} = \frac{24}{30} = 0.80 = 80\%
\]

Calculate the marginal probability of liking pizza

\[
P(\text{Likes Pizza}) = \frac{\text{Total Likes Pizza}}{\text{Grand Total}} = \frac{40}{50} = 0.80 = 80\%
\]

Determine independence of the events

\[

$$\begin{aligned} &P(\text{Likes Pizza} \mid \text{Female}) = P(\text{Likes Pizza}) = 80\% \\ &\text{Since the probabilities are equal, the events are independent.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>48%; Yes</mcq-option>
<mcq-correct>80%; Yes</mcq-correct>
<mcq-option>48%; No</mcq-option>
<mcq-option>60%; No</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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"step_cot",
"multiple_choice_formatter",
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"current_concepts": [
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"Probability Formulas",
"Two-Way Frequency Tables",
"Conditional Probability"
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}
</pre_analysis>

<reasoning>

Calculate the conditional probability of a female liking pizza

\[
P(\text{Likes Pizza} \mid \text{Female}) = \frac{\text{Female and Likes Pizza}}{\text{Total Female}} = \frac{24}{30} = 0.80 = 80\%
\]

Calculate the marginal probability of liking pizza

\[
P(\text{Likes Pizza}) = \frac{\text{Total Likes Pizza}}{\text{Grand Total}} = \frac{40}{50} = 0.80 = 80\%
\]

Determine independence of the events

\[

$$\begin{aligned} &P(\text{Likes Pizza} \mid \text{Female}) = P(\text{Likes Pizza}) = 80\% \\ &\text{Since the probabilities are equal, the events are independent.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>48%; Yes</mcq-option>
<mcq-correct>80%; Yes</mcq-correct>
<mcq-option>48%; No</mcq-option>
<mcq-option>60%; No</mcq-option>
</answer>

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