Question

in 2023, mr. hummels ap stats class was asked if they were born in cky and what their favorite season is. the results are summarized below in a two - way table.

were	what is your favorite season?
no	9	1	6	4	20
yes	10	6	10	12	38
grand	19	7	16	16	58

one of the students from this group is randomly chosen. which is p(favorite is the favorite season | born in cky)?
a. 10/38
b. 10/16
c. none of these
d. a and b
e. 5/16

Explanation:

Step1: Identify the relevant groups

We need the number of students born in the US (Yes) and whose favorite season is Fall. From the table, the number of students with "Yes" (born in US) and "Fall" as favorite season is 10. The total number of students born in the US (Yes) is 38 (from the "Grand" column of the "Yes" row). Wait, no, wait. Wait, the question is "P(favorite season is Fall | born in US)". So this is a conditional probability. The formula for conditional probability is \( P(A|B) = \frac{P(A \cap B)}{P(B)} \), which is equivalent to \( \frac{\text{Number of students with } A \text{ and } B}{\text{Number of students with } B} \). Here, \( A \) is "favorite season is Fall" and \( B \) is "born in US".

From the table, the number of students born in US (Yes) and favorite season Fall is 10 (row "Yes", column "Fall"). The total number of students born in US (Yes) is the grand total of the "Yes" row, which is 38? Wait, no, looking at the table again: the "Yes" row has Fall:10, Spring:6, Summ:10, Winter:12, Grand: 10+6+10+12=38. So the number of students born in US is 38, and the number of those born in US with favorite season Fall is 10. So the probability is \( \frac{10}{38} \)? Wait, but the options include 10/38? Wait, option A is 10/38? Wait, the options: A is 10/38, B is 10/16, etc. Wait, maybe I misread the table. Wait, the "Grand" column for "Yes" is 38? Let me check the table again. The table:

Rows: No, Yes, Grand.

Columns: Fall, Spring, Summ, Winter, Grand.

No row: Fall:9, Spring:1, Summ:6, Winter:4, Grand:9+1+6+4=20.

Yes row: Fall:10, Spring:6, Summ:10, Winter:12, Grand:10+6+10+12=38.

Grand row: Fall:19, Spring:7, Summ:16, Winter:16, Grand:20+38=58.

So the number of students born in US (Yes) is 38, and the number of those born in US with favorite season Fall is 10. So \( P(\text{Fall} | \text{Yes}) = \frac{10}{38} \). Looking at the options, option A is 10/38. So that's the correct one.

Step2: Verify the calculation

Conditional probability formula: \( P(\text{Fall} | \text{Born in US}) = \frac{\text{Number of students born in US and favorite season Fall}}{\text{Number of students born in US}} = \frac{10}{38} \). So option A is 10/38, which matches.

Answer:

A. 10/38