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A burrito restaurant used the given table to keep track of the numbers of different types of burritos that were sold and the kinds of beans that people requested.

Consider the following events:
A. The burrito is a chicken burrito.
B. The burrito is a carne asada burrito.
C. The customer requested black beans.
D. The customer requested pinto beans.

\begin{tabular}{|c|c|c|c|c|c|}
\hline & Fish & Chicken & \begin{tabular}{c}
Carne \\
Asada
\end{tabular} & Vegetarian & Total \\
\hline Black Beans & a & 37 & 5 & 1 & 45 \\
\hline Pinto Beans & 10 & 30 & 24 & 8 & 72 \\
\hline No Beans & 36 & 16 & 51 & 20 & 123 \\
\hline Total & 48 & 83 & 80 & 29 & 240 \\
\hline
\end{tabular}

Which two events are independent?
A. A and C
B. A and D

Sagot :

GinnyAnswer
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.
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