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Sagot :
To determine the independence of two events, we compare the joint probability of the events with the product of their individual (marginal) probabilities. Events [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] are independent if and only if [tex]\( P(X \ \text{and} \ Y) = P(X) \times P(Y) \)[/tex].
Given the data from the contingency table and our recognized events:
- [tex]\(A\)[/tex]: The burrito is a chicken burrito.
- [tex]\(B\)[/tex]: The burrito is a carne asada burrito.
- [tex]\(C\)[/tex]: The customer requested black beans.
- [tex]\(D\)[/tex]: The customer requested pinto beans.
### Step-by-Step Solution:
1. Calculate the total number of samples:
[tex]\[ \text{Total count} = 240 \][/tex]
2. Calculate the marginal probabilities:
[tex]\[ P(A) = \frac{83}{240} \][/tex]
[tex]\[ P(B) = \frac{80}{240} \][/tex]
[tex]\[ P(C) = \frac{45}{240} \][/tex]
[tex]\[ P(D) = \frac{72}{240} \][/tex]
3. Calculate the joint probabilities:
[tex]\[ P(A \ \text{and} \ C) = \frac{37}{240} \][/tex]
[tex]\[ P(A \ \text{and} \ D) = \frac{30}{240} \][/tex]
[tex]\[ P(B \ \text{and} \ C) = \frac{5}{240} \][/tex]
[tex]\[ P(B \ \text{and} \ D) = \frac{24}{240} \][/tex]
4. Compare the joint probability with the product of marginal probabilities:
a. For [tex]\( A \ \text{and} \ C \)[/tex]:
[tex]\[ P(A \ \text{and} \ C) \neq P(A) \times P(C) \][/tex]
[tex]\[ \frac{37}{240} \neq \frac{83}{240} \times \frac{45}{240} \][/tex]
Therefore, [tex]\( A \ \text{and} \ C \)[/tex] are not independent.
b. For [tex]\( A \ \text{and} \ D \)[/tex]:
[tex]\[ P(A \ \text{and} \ D) \neq P(A) \times P(D) \][/tex]
[tex]\[ \frac{30}{240} \neq \frac{83}{240} \times \frac{72}{240} \][/tex]
Therefore, [tex]\( A \ \text{and} \ D \)[/tex] are not independent.
c. For [tex]\( B \ \text{and} \ C \)[/tex]:
[tex]\[ P(B \ \text{and} \ C) \neq P(B) \times P(C) \][/tex]
[tex]\[ \frac{5}{240} \neq \frac{80}{240} \times \frac{45}{240} \][/tex]
Therefore, [tex]\( B \ \text{and} \ C \)[/tex] are not independent.
d. For [tex]\( B \ \text{and} \ D \)[/tex]:
[tex]\[ P(B \ \text{and} \ D) \neq P(B) \times P(D) \][/tex]
[tex]\[ \frac{24}{240} \neq \frac{80}{240} \times \frac{72}{240} \][/tex]
Therefore, [tex]\( B \ \text{and} \ D \)[/tex] are not independent.
After examining all pairs, we can confirm that none of the events [tex]\(A\)[/tex] and [tex]\(C\)[/tex], [tex]\(A\)[/tex] and [tex]\(D\)[/tex], [tex]\(B\)[/tex] and [tex]\(C\)[/tex], [tex]\(B\)[/tex] and [tex]\(D\)[/tex] are independent.
Thus, the answer is:
None of the given pairs of events are independent.
Given the data from the contingency table and our recognized events:
- [tex]\(A\)[/tex]: The burrito is a chicken burrito.
- [tex]\(B\)[/tex]: The burrito is a carne asada burrito.
- [tex]\(C\)[/tex]: The customer requested black beans.
- [tex]\(D\)[/tex]: The customer requested pinto beans.
### Step-by-Step Solution:
1. Calculate the total number of samples:
[tex]\[ \text{Total count} = 240 \][/tex]
2. Calculate the marginal probabilities:
[tex]\[ P(A) = \frac{83}{240} \][/tex]
[tex]\[ P(B) = \frac{80}{240} \][/tex]
[tex]\[ P(C) = \frac{45}{240} \][/tex]
[tex]\[ P(D) = \frac{72}{240} \][/tex]
3. Calculate the joint probabilities:
[tex]\[ P(A \ \text{and} \ C) = \frac{37}{240} \][/tex]
[tex]\[ P(A \ \text{and} \ D) = \frac{30}{240} \][/tex]
[tex]\[ P(B \ \text{and} \ C) = \frac{5}{240} \][/tex]
[tex]\[ P(B \ \text{and} \ D) = \frac{24}{240} \][/tex]
4. Compare the joint probability with the product of marginal probabilities:
a. For [tex]\( A \ \text{and} \ C \)[/tex]:
[tex]\[ P(A \ \text{and} \ C) \neq P(A) \times P(C) \][/tex]
[tex]\[ \frac{37}{240} \neq \frac{83}{240} \times \frac{45}{240} \][/tex]
Therefore, [tex]\( A \ \text{and} \ C \)[/tex] are not independent.
b. For [tex]\( A \ \text{and} \ D \)[/tex]:
[tex]\[ P(A \ \text{and} \ D) \neq P(A) \times P(D) \][/tex]
[tex]\[ \frac{30}{240} \neq \frac{83}{240} \times \frac{72}{240} \][/tex]
Therefore, [tex]\( A \ \text{and} \ D \)[/tex] are not independent.
c. For [tex]\( B \ \text{and} \ C \)[/tex]:
[tex]\[ P(B \ \text{and} \ C) \neq P(B) \times P(C) \][/tex]
[tex]\[ \frac{5}{240} \neq \frac{80}{240} \times \frac{45}{240} \][/tex]
Therefore, [tex]\( B \ \text{and} \ C \)[/tex] are not independent.
d. For [tex]\( B \ \text{and} \ D \)[/tex]:
[tex]\[ P(B \ \text{and} \ D) \neq P(B) \times P(D) \][/tex]
[tex]\[ \frac{24}{240} \neq \frac{80}{240} \times \frac{72}{240} \][/tex]
Therefore, [tex]\( B \ \text{and} \ D \)[/tex] are not independent.
After examining all pairs, we can confirm that none of the events [tex]\(A\)[/tex] and [tex]\(C\)[/tex], [tex]\(A\)[/tex] and [tex]\(D\)[/tex], [tex]\(B\)[/tex] and [tex]\(C\)[/tex], [tex]\(B\)[/tex] and [tex]\(D\)[/tex] are independent.
Thus, the answer is:
None of the given pairs of events are independent.
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