Let's analyze the given events and data step-by-step to determine the independence of the events A and C, and A and D.
1. Total Burritos Sold:
[tex]\[
\text{Total } = 240
\][/tex]
2. Event A: The burrito is a chicken burrito.
- There are 83 chicken burritos sold.
- Probability [tex]\( P(A) \)[/tex]:
[tex]\[
P(A) = \frac{83}{240} = 0.345833
\][/tex]
3. Event C: The customer requested black beans.
- There are 45 black beans requested.
- Probability [tex]\( P(C) \)[/tex]:
[tex]\[
P(C) = \frac{45}{240} = 0.1875
\][/tex]
4. Combined Event A and C: The burrito is a chicken burrito and the customer requested black beans.
- There are 37 chicken burritos with black beans.
- Probability [tex]\( P(A \cap C) \)[/tex]:
[tex]\[
P(A \cap C) = \frac{37}{240} = 0.154167
\][/tex]
To check for independence of A and C, we compare [tex]\( P(A \cap C) \)[/tex] with [tex]\( P(A) \times P(C) \)[/tex]:
[tex]\[
\text{If } P(A \cap C) = P(A) \times P(C), \text{ then A and C are independent.}
\][/tex]
Calculation:
[tex]\[
P(A) \times P(C) = 0.345833 \times 0.1875 = 0.064167
\][/tex]
Since:
[tex]\[
P(A \cap C) \neq P(A) \times P(C)
\][/tex]
Events A and C are not independent.
5. Event D: The customer requested pinto beans.
- There are 72 pinto beans requested.
- Probability [tex]\( P(D) \)[/tex]:
[tex]\[
P(D) = \frac{72}{240} = 0.3
\][/tex]
6. Combined Event A and D: The burrito is a chicken burrito and the customer requested pinto beans.
- There are 30 chicken burritos with pinto beans.
- Probability [tex]\( P(A \cap D) \)[/tex]:
[tex]\[
P(A \cap D) = \frac{30}{240} = 0.125
\][/tex]
To check for independence of A and D, we compare [tex]\( P(A \cap D) \)[/tex] with [tex]\( P(A) \times P(D) \)[/tex]:
[tex]\[
\text{If } P(A \cap D) = P(A) \times P(D), \text{ then A and D are independent.}
\][/tex]
Calculation:
[tex]\[
P(A) \times P(D) = 0.345833 \times 0.3 = 0.103750
\][/tex]
Since:
[tex]\[
P(A \cap D) \neq P(A) \times P(D)
\][/tex]
Events A and D are not independent.
---
In summary:
- Events A and C are not independent.
- Events A and D are not independent either.
