Consider the Venn diagram below. The numbers in the regions of the circle indicate the number of items that belong to that region.
(2 intersecting circles A & B, where A part is 50, B part is 110, and the intersecting part is 40)
Determine:
n(A)
n(B)
P(A)
P(B)
P(A|B)
P(B|A)
counting all the numbers in A I see n(A) = 90
n(B)=150
total is 200 , P(A) = 90/200 = 9/20
P(B) = 150/200 = 3/4
P(A|B) is a conditional probability and is defined as
P(A∩B/P(B)
= (40/200) / (3/4) = 4/15
find P(B|A) the same way
missed a bracket ...
P(A|B) is a conditional probability and is defined as
P(A∩B) / P(B)
To determine the values of n(A), n(B), P(A), P(B), P(A|B), and P(B|A) based on the given Venn diagram, follow these steps:
1. n(A) represents the number of items in set A. According to the diagram, the number of items in the A part is 50. Therefore, n(A) = 50.
2. n(B) represents the number of items in set B. From the diagram, the number of items in the B part is 110. Hence, n(B) = 110.
3. P(A) represents the probability of an event A occurring. To find P(A), divide the number of items in A (n(A)) by the total number of items in the sample space (n(A) + n(B) - intersecting part). In this case, the total number of items in the sample space is (50 + 110 - 40) = 120. Thus, P(A) = 50/120 = 5/12 or approximately 0.4167.
4. P(B) represents the probability of an event B occurring. Similar to P(A), divide the number of items in B (n(B)) by the total number of items in the sample space. In this case, P(B) = 110/120 = 11/12 or approximately 0.9167.
5. P(A|B) represents the probability of event A occurring given that event B has already occurred. To calculate it, divide the number of items in the intersecting part of A and B by the number of items in B. In the diagram, the intersecting part has 40 items, and B has 110. Therefore, P(A|B) = 40/110 = 4/11 or approximately 0.3636.
6. P(B|A) represents the probability of event B occurring given that event A has already occurred. To find it, divide the number of items in the intersecting part of A and B by the number of items in A. In this case, P(B|A) = 40/50 = 4/5, which is equal to 0.8 or 80%.
In summary:
n(A) = 50
n(B) = 110
P(A) = 5/12 or approximately 0.4167
P(B) = 11/12 or approximately 0.9167
P(A|B) = 4/11 or approximately 0.3636
P(B|A) = 4/5 or 0.8 (or 80%).