Question 1123692: J, K, and L are events in sample space S.
Pr(J)=0.29
Pr(K)=0.23
Pr(L)=0.39
Pr(J∩K)=0.13
Pr(J′∩L′)=0.51
Pr(K′∩L)=0.28
What is Pr(J∪K)?
What is Pr(J∩L)?
What is Pr(K∩L′)?
Enter your answers as whole numbers or decimals.
Answer by greenestamps(13203)   (Show Source):
You can put this solution on YOUR website!
Refer to the above diagram for the discussion below.
I will use whole numbers (percentages) to avoid having to write all the decimal numbers.
The givens are:
(1) Pr(J) = 29 (a+b+c+e)
(2) Pr(K) = 23 (a+c+d+f)
(3) Pr(L) = 39 (a+b+d+g)
(4) Pr(JK) = 13 (a+c)
(5) Pr(J'L') = 51 (we can't really use this, because it includes regions outside of the union of J, K, and L)
(6) Pr(K'L) = 28 (b+g)
It turns out we need to introduce two variables:
(7) a = x
(8) b = y
Then we can do some calculations to put expressions involving x and y in regions c, d, e, f, and g.
(9) g = 28-y (from (6) and (8)
(10) d = 11-x (from 3, 7, 8, and 9)
(11) c = 13-x (from 4 and 7)
(12) e = 16-y (from 1, 7, 8, and 11)
(13) f = x-1 (from 2, 7, 10, and 11)
And now we can answer the questions that are asked.
What is Pr(J∪K)? Answer: a+b+c+d+e+f = 39; so Pr(J∪K) = 0.39.
What is Pr(J∩L)? Answer: a+b = x+y.
It is not possible, with the given information, to find a numerical value of Pr(J∩L). (10) and (13) above restrict the possible values of x to between 1 and 11; (12) restricts the possible values of y to between 0 and 16. But any combination of those values of x and y yields a solution to the problem.
What is Pr(K∩L′)? Answer: c+f = 12; so Pr(K∩L′) = 0.12.
|
|
|
