Question

a survey was completed at a high school with 745 students to determine their favorite time of year. the results are below.

favorite time of the year

	winter	spring	summer	fall
sophomore	20	28	96	52
junior	42	26	52	48
senior	30	25	42	34

1. what is the probability a student prefers winter given they are a freshman?
2. find p(junior | fall).
3. find p(summer | senior).
4. if a student is chosen at random, what is the probability it will be a student who is a sophomore or a student who likes spring.

Explanation:

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<reasoning>

Calculate row and column totals

\[

$$\begin{aligned} &\text{Total Freshman} = 25 + 35 + 88 + 102 = 250\\ &\text{Total Sophomore} = 20 + 28 + 96 + 52 = 196\\ &\text{Total Junior} = 42 + 26 + 52 + 48 = 168\\ &\text{Total Senior} = 30 + 25 + 42 + 34 = 131\\ &\text{Total Winter} = 25 + 20 + 42 + 30 = 117\\ &\text{Total Spring} = 35 + 28 + 26 + 25 = 114\\ &\text{Total Summer} = 88 + 96 + 52 + 42 = 278\\ &\text{Total Fall} = 102 + 52 + 48 + 34 = 236\\ &\text{Grand Total} = 250 + 196 + 168 + 131 = 745 \end{aligned}$$

\]

Solve conditional probabilities (Questions 1, 2, and 3)

\[

$$\begin{aligned} &P(\text{Winter} \mid \text{Freshman}) = \frac{\text{Freshman and Winter}}{\text{Total Freshman}} = \frac{25}{250} = \frac{1}{10} = 0.1\\ &P(\text{Junior} \mid \text{Fall}) = \frac{\text{Junior and Fall}}{\text{Total Fall}} = \frac{48}{236} = \frac{12}{59} \approx 0.2034\\ &P(\text{Summer} \mid \text{Senior}) = \frac{\text{Senior and Summer}}{\text{Total Senior}} = \frac{42}{131} \approx 0.3206 \end{aligned}$$

\]

Solve union probability (Question 4)

\[

$$\begin{aligned} &P(\text{Sophomore} \cup \text{Spring}) = P(\text{Sophomore}) + P(\text{Spring}) - P(\text{Sophomore} \cap \text{Spring})\\ &P(\text{Sophomore} \cup \text{Spring}) = \frac{196}{745} + \frac{114}{745} - \frac{28}{745} = \frac{282}{745} \approx 0.3785 \end{aligned}$$

\]
</reasoning>

<answer>

No.	Answer
2	\(\frac{48}{236} = \frac{12}{59}\) (or \(\approx 0.203\))
3	\(\frac{42}{131}\) (or \(\approx 0.321\))
4	\(\frac{282}{745}\) (or \(\approx 0.379\))

</answer>

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Answer:

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<reasoning>

Calculate row and column totals

\[

$$\begin{aligned} &\text{Total Freshman} = 25 + 35 + 88 + 102 = 250\\ &\text{Total Sophomore} = 20 + 28 + 96 + 52 = 196\\ &\text{Total Junior} = 42 + 26 + 52 + 48 = 168\\ &\text{Total Senior} = 30 + 25 + 42 + 34 = 131\\ &\text{Total Winter} = 25 + 20 + 42 + 30 = 117\\ &\text{Total Spring} = 35 + 28 + 26 + 25 = 114\\ &\text{Total Summer} = 88 + 96 + 52 + 42 = 278\\ &\text{Total Fall} = 102 + 52 + 48 + 34 = 236\\ &\text{Grand Total} = 250 + 196 + 168 + 131 = 745 \end{aligned}$$

\]

Solve conditional probabilities (Questions 1, 2, and 3)

\[

$$\begin{aligned} &P(\text{Winter} \mid \text{Freshman}) = \frac{\text{Freshman and Winter}}{\text{Total Freshman}} = \frac{25}{250} = \frac{1}{10} = 0.1\\ &P(\text{Junior} \mid \text{Fall}) = \frac{\text{Junior and Fall}}{\text{Total Fall}} = \frac{48}{236} = \frac{12}{59} \approx 0.2034\\ &P(\text{Summer} \mid \text{Senior}) = \frac{\text{Senior and Summer}}{\text{Total Senior}} = \frac{42}{131} \approx 0.3206 \end{aligned}$$

\]

Solve union probability (Question 4)

\[

$$\begin{aligned} &P(\text{Sophomore} \cup \text{Spring}) = P(\text{Sophomore}) + P(\text{Spring}) - P(\text{Sophomore} \cap \text{Spring})\\ &P(\text{Sophomore} \cup \text{Spring}) = \frac{196}{745} + \frac{114}{745} - \frac{28}{745} = \frac{282}{745} \approx 0.3785 \end{aligned}$$

\]
</reasoning>

<answer>

No.	Answer
2	\(\frac{48}{236} = \frac{12}{59}\) (or \(\approx 0.203\))
3	\(\frac{42}{131}\) (or \(\approx 0.321\))
4	\(\frac{282}{745}\) (or \(\approx 0.379\))

</answer>

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