To find the probability that a randomly chosen customer has purchased either a tee shirt or an item that is small, we need to use the principle of inclusion-exclusion. This principle helps us compute the probability of either of two events happening by accounting for their overlap.
First, let's summarize the given data based on the table:
- Tee Shirt:
- Small: 71
- Medium: 97
- Large: 31
- Long Sleeve Shirt:
- Small: 38
- Medium: 41
- Large: 11
- Jacket:
- Small: 2
- Medium: 23
- Large: 0
### Step 1: Total Purchases by Category
1. Total Tee Shirts:
[tex]\[
71 \text{ (small)} + 97 \text{ (medium)} + 31 \text{ (large)} = 199
\][/tex]
2. Total Small Items:
[tex]\[
71 \text{ (tee shirt small)} + 38 \text{ (long sleeve small)} + 2 \text{ (jacket small)} = 111
\][/tex]
3. Total Purchases Overall:
[tex]\[
(71 + 97 + 31) \text{ (tee shirts)} + (38 + 41 + 11) \text{ (long sleeves)} + (2 + 23 + 0) \text{ (jackets)} = 314
\][/tex]
### Step 2: Probabilities
1. Probability of Choosing a Tee Shirt (P(Tee Shirt)):
[tex]\[
P(\text{Tee Shirt}) = \frac{\text{Total Tee Shirts}}{\text{Total Purchases}} = \frac{199}{314}
\][/tex]
2. Probability of Choosing a Small Item (P(Small)):
[tex]\[
P(\text{Small}) = \frac{\text{Total Small Items}}{\text{Total Purchases}} = \frac{111}{314}
\][/tex]
3. Probability of Choosing a Tee Shirt and a Small Item (P(Tee Shirt AND Small)):
[tex]\[
P(\text{Tee Shirt AND Small}) = \frac{71}{314} \quad \text{(since 71 customers bought small tee shirts)}
\][/tex]
### Step 3: Using Inclusion-Exclusion Principle
According to the principle of inclusion-exclusion, the probability of choosing a tee shirt or a small item (P(Tee Shirt OR Small)) is:
[tex]\[
P(\text{Tee Shirt OR Small}) = P(\text{Tee Shirt}) + P(\text{Small}) - P(\text{Tee Shirt AND Small})
\][/tex]
### Step 4: Substituting the Values
[tex]\[
P(\text{Tee Shirt OR Small}) = \frac{199}{314} + \frac{111}{314} - \frac{71}{314}
\][/tex]
Combining these fractions directly:
[tex]\[
P(\text{Tee Shirt OR Small}) = \frac{199 + 111 - 71}{314} = \frac{239}{314}
\][/tex]
Hence, the probability that a randomly chosen customer has purchased either a tee shirt or a small item is:
[tex]\[
P(\text{Tee Shirt OR Small}) = \frac{239}{314}
\][/tex]
