Question

a survey asked about the number of people who eat breakfast almost every day (b) and the number of people who buy cereal at least once a month (c). the results of the survey are shown in the venn diagram. given that a randomly chosen person eats breakfast almost everyday, what is the probability that the person also buys cereal at least once a month? options: \\(\frac{11}{64}\\), \\(\frac{11}{53}\\), \\(\frac{53}{64}\\), \\(\frac{53}{57}\\)

Explanation:

Step1: Identify total for event B

To find the total number of people who eat breakfast almost every day (event B), we add the number of people only in B and the number in the intersection of B and C. So, \( 11 + 53 = 64 \).

Step2: Identify favorable outcomes

The number of people who eat breakfast almost every day and also buy cereal (favorable outcomes) is the number in the intersection, which is 53.

Step3: Calculate conditional probability

The formula for conditional probability \( P(C|B) \) is \( \frac{P(C \cap B)}{P(B)} \), which is equivalent to \( \frac{\text{Number of } C \cap B}{\text{Number of } B} \). Substituting the values, we get \( \frac{53}{64} \)? Wait, no, wait. Wait, the total for B is 11 (only B) + 53 (both) = 64? Wait, no, wait the universal set: Wait, the Venn diagram has B (11 only, 53 both), C (4 only, 53 both), and outside both is 23. Wait, but the conditional probability is given that the person eats breakfast almost every day (B), so we consider the sample space as B. So the number of people in B is 11 (only B) + 53 (both B and C) = 64? Wait, no, wait the question is "given that a randomly chosen person eats breakfast almost everyday, what is the probability that the person also buys cereal at least once a month?" So that's \( P(C|B) = \frac{n(C \cap B)}{n(B)} \). \( n(C \cap B) = 53 \), \( n(B) = 11 + 53 = 64 \)? Wait, but the options have \( \frac{53}{57} \)? Wait, maybe I miscalculated n(B). Wait, maybe the total number of people who eat breakfast almost every day is 11 (only B) + 53 (both) = 64? But the universal set: Wait, the outside is 23, but maybe the total number of people is 11 + 53 + 4 + 23 = 91? But no, conditional probability is within B. Wait, no, let's re-express:

Wait, the Venn diagram: Circle B has 11 (only B) and 53 (overlap with C). Circle C has 4 (only C) and 53 (overlap with B). Outside both is 23. So the number of people who eat breakfast almost every day (B) is 11 + 53 = 64? But the option \( \frac{53}{57} \): 53 + 11 + 4 + 23 = 91? No, 11 + 53 + 4 + 23 = 91. Wait, maybe the total number of people who eat breakfast almost every day is 11 + 53 = 64? But 53 + 11 = 64, and 53 + 4 + 23 = 80? No, 53 + 4 is 57, plus 23 is 80? Wait, I think I made a mistake. Wait, the conditional probability is \( P(C|B) = \frac{n(C \cap B)}{n(B)} \). \( n(C \cap B) = 53 \). \( n(B) = 11 + 53 = 64 \)? But the option \( \frac{53}{57} \): 57 would be 11 + 53 + 4 - no, wait maybe the total number of people who eat breakfast almost every day is 11 (only B) + 53 (both) = 64? But the option \( \frac{53}{57} \): 57 is 11 + 53 + 4 - no, that's C. Wait, no, maybe the total number of people who eat breakfast almost every day is 11 + 53 = 64, but the option \( \frac{53}{57} \) is wrong? Wait, no, maybe I misread the Venn diagram. Wait, the numbers: B has 11 (only B), overlap is 53, C has 4 (only C), and outside is 23. So the number of people who eat breakfast almost every day (B) is 11 + 53 = 64. The number of people who eat breakfast and buy cereal is 53. So \( P(C|B) = \frac{53}{64} \), but that's an option (third option). Wait, but the fourth option is \( \frac{53}{57} \). Wait, maybe the total number of people who eat breakfast almost every day is 11 + 53 = 64? Wait, 11 + 53 = 64, 53 + 4 + 23 = 80, 11 + 53 + 4 + 23 = 91. No, maybe the question is using the total number of people who eat breakfast as 11 + 53 = 64, so \( \frac{53}{64} \) is option C. Wait, but let's check again.

Wait, the formula for conditional probability is \( P(C|B) = \frac{n(C \cap B)}{n(B)} \). \( n(C \cap B) = 53 \), \( n(B) = 11 + 53 = 64 \). So \( \frac{53}{64} \), which is the third option.

Answer:

\(\frac{53}{64}\) (Option: \(\frac{53}{64}\))