I can't imagine how to draw a Venn diagram for this problem.
First, probability that P is correct: .John has 5 items named .
Each item is placed in a separate box and sealed.
He then addresses the boxes at random, one each to
1) Find the probability that the item is in the correct box
and item is in the wrong box. (3/20)
Then probability that Q is incorrect: .
Bayes' Theorem: . .2) Find the probability that the item is in the correct box,
given that the item is in the wrong box. (3/16)
We found the numerator in part (1): .
The denominator is: .
Substitute into : .
This is a bit trickier . . .3) Find the probability that both and are in the wrong boxes. (13/20)
There are two scenarios:
. . (1) goes to
. . (2) does not go to
(1) goes to
. . .We have: .
. . The other four items can have any order
. . and will be in the wrong box.
. . .Hence: .
(2) does not go to
. . . . . . . . . . . .
. . .
. . . can go to any of the last three addresses: .
. . .Then can go to three of the remaining four addresses: .
. . . Hence: .