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)
What āPā stands for?
P stands for probability
To determine the values you asked for, follow the explanations below:
n(A) represents the number of items that belong to set A. Looking at the Venn diagram, the value given for the region of circle A is 50, so n(A) = 50.
n(B) represents the number of items that belong to set B. Looking at the Venn diagram, the value given for the region of circle B is 110, so n(B) = 110.
P(A) represents the probability of an event occurring in set A. To find P(A), divide the number of items in set A by the total number of items in the sample space. In this case, the total number of items can be determined by adding the values of the individual regions of circle A and circle B. So, P(A) = n(A) / (n(A) + n(B) - intersecting part) = 50 / (50 + 110 - 40) = 50 / 120 = 5/12 = 0.4167 (rounded to four decimal places).
P(B) represents the probability of an event occurring in set B. To find P(B), divide the number of items in set B by the total number of items in the sample space. So, P(B) = n(B) / (n(A) + n(B) - intersecting part) = 110 / (50 + 110 - 40) = 110 / 120 = 11/12 = 0.9167 (rounded to four decimal places).
P(A|B) represents the probability of an event occurring in set A given that it's already known that event B has occurred. To find P(A|B), divide the number of items in the intersecting part of A and B by the number of items in set B. In this case, the intersection part is given as 40, and n(B) is given as 110. So, P(A|B) = intersecting part / n(B) = 40 / 110 = 4/11 = 0.3636 (rounded to four decimal places).
P(B|A) represents the probability of an event occurring in set B given that it's already known that event A has occurred. To find P(B|A), divide the number of items in the intersecting part of A and B by the number of items in set A. So, P(B|A) = intersecting part / n(A) = 40 / 50 = 4/5 = 0.8.
So, the values are:
n(A) = 50
n(B) = 110
P(A) = 0.4167
P(B) = 0.9167
P(A|B) = 0.3636
P(B|A) = 0.8