Prepare a Venn diagram giving the probabilities for each distinct area; then answer the questions posed. A survey of 168 boys aged 6 to 12 revealed the following information: 74 of them liked chocolate cake, 89 of them liked golden cake and 23 liked both.
A) What is the probability that a boy liked golden cake, given that he liked chocolate cake?
B) What is the probability that a boy didn't like golden cake given that he liked chocolate cake?
C) Are liking chocolate cake and not liking golden cake statistically independent? Give mathematical proof.
Any help regarding this question would be greatly appreciated as I find it quite confusing how to calculate the areas of the Venn Diagram and the following questions. I understand I am using formulas such as P(A|B) for the questions, but I am still very, very confused by the question.
To solve this problem, we can use the concept of conditional probability and the information provided in the survey. Let's begin by constructing a Venn diagram to visualize the information.
First, we know that there are 168 boys in the survey. Let's denote the probability of liking chocolate cake as P(C) and the probability of liking golden cake as P(G). We also know that 23 boys liked both chocolate cake and golden cake. Let's denote this overlapping region as P(C ∩ G).
Now, let's calculate the probabilities for each distinct region in the Venn diagram:
1. The number of boys who liked chocolate cake only (not golden cake) is given by 74 - 23 = 51. Denote this region as P(C) - P(C ∩ G).
2. The number of boys who liked golden cake only (not chocolate cake) is given by 89 - 23 = 66. Denote this region as P(G) - P(C ∩ G).
3. The number of boys who liked both chocolate cake and golden cake is already given as 23. Denote this region as P(C ∩ G).
Now, let's calculate the probabilities for each of these regions:
1. P(C) = 51 / 168, since there are 51 boys who liked chocolate cake only out of the total 168 boys.
2. P(G) = 66 / 168, since there are 66 boys who liked golden cake only out of the total 168 boys.
3. P(C ∩ G) = 23 / 168, since there are 23 boys who liked both chocolate cake and golden cake out of the total of 168 boys.
A) To find the probability that a boy liked golden cake, given that he liked chocolate cake, we need to calculate P(G|C). This can be done using the formula for conditional probability:
P(G|C) = P(C ∩ G) / P(C)
Plugging in the values we calculated:
P(G|C) = (23 / 168) / (51 / 168)
= 23 / 51
Therefore, the probability that a boy liked golden cake, given that he liked chocolate cake, is 23/51.
B) To find the probability that a boy didn't like golden cake, given that he liked chocolate cake, we need to calculate P(~G|C). The complement of liking golden cake is not liking golden cake (~G).
P(~G|C) = 1 - P(G|C) [using the fact that P(~G) = 1 - P(G)]
Plugging in the value we calculated for P(G|C):
P(~G|C) = 1 - (23 / 51)
= 28 / 51
Therefore, the probability that a boy didn't like golden cake given that he liked chocolate cake is 28/51.
C) To determine if liking chocolate cake and not liking golden cake are statistically independent, we need to check if P(C ∩ ~G) = P(C) * P(~G). If they are equal, then the events are independent. Let's calculate the probabilities involved:
P(C ∩ ~G) = (P(C) - P(C ∩ G)) = (51 / 168) - (23 / 168) = 28 / 168 = 1 / 6
P(C) * P(~G) = (51 / 168) * (66 / 168) = 33 / 168 = 11 / 56
Since P(C ∩ ~G) ≠ P(C) * P(~G), we can conclude that liking chocolate cake and not liking golden cake are not statistically independent.
I hope this explanation helps in understanding how to calculate probabilities using a Venn diagram and solve the given questions. If you have any further queries, feel free to ask.