Question

data gathered on the shopping patterns during the months of april and may of high school students from peanut village revealed the following. 38% of students purchased a new pair of shorts (call this event h), 15% of students purchased a new pair of sunglasses (call this event g), and 6% of students purchased both a pair of shorts and a pair of sunglasses.

1. find the probability that a student purchased a pair of sunglasses given that you know they purchased a pair of shorts.

$p(g|h) =$

1. find the probability that a student purchased a pair of shorts or purchased a new pair of sunglasses.

$p(h \text{ or } g) =$

1. given that you know a student has purchased at least one of the items, what is the probability that they purchased only one of the items?
2. are the two events, h and g, independent of one another? why or why not?

(handwritten notes and a venn diagram are present: including \41%\, \9% 6% 32%\ in the venn diagram, formulas like $p(a|b)=\frac{p(a\cap b)}{p(b)}$, \38 - 6 = 32\, and other handwritten calculations.)

Explanation:

Question 9

Step1: Recall Conditional Probability Formula

The formula for conditional probability is \( P(G|H) = \frac{P(G \cap H)}{P(H)} \).

Step2: Identify Given Probabilities

We know \( P(H) = 0.38 \) (38% of students purchased shorts) and \( P(G \cap H) = 0.06 \) (6% of students purchased both).

Step3: Substitute into Formula

Substitute the values into the formula: \( P(G|H) = \frac{0.06}{0.38} \approx \frac{3}{19} \approx 0.1579 \) (or 15.79%).

Step1: Recall Probability of Union Formula

The formula for \( P(H \text{ or } G) \) is \( P(H) + P(G) - P(H \cap G) \).

Step2: Identify Given Probabilities

We have \( P(H) = 0.38 \), \( P(G) = 0.15 \), and \( P(H \cap G) = 0.06 \).

Step3: Substitute into Formula

Substitute the values: \( P(H \text{ or } G) = 0.38 + 0.15 - 0.06 = 0.47 \) (or 47%).

Step1: Find Probability of Only One Item

First, find the number of students who purchased only shorts: \( P(\text{only } H) = P(H) - P(H \cap G) = 0.38 - 0.06 = 0.32 \).
Then, find the number of students who purchased only sunglasses: \( P(\text{only } G) = P(G) - P(H \cap G) = 0.15 - 0.06 = 0.09 \).
The probability of purchasing only one item is \( P(\text{only } H) + P(\text{only } G) = 0.32 + 0.09 = 0.41 \).

Step2: Find Probability of At Least One Item

From Question 10, \( P(H \text{ or } G) = 0.47 \) (this is the probability of purchasing at least one item).

Step3: Apply Conditional Probability

The conditional probability is \( \frac{P(\text{only one item})}{P(H \text{ or } G)} = \frac{0.41}{0.47} \approx 0.8723 \) (or 87.23%).

Answer:

\( \frac{3}{19} \) (or approximately \( 0.1579 \) or 15.79%)

Question 10