Now, the ball drawn may be black or non-black (any of the remaining 5 different colours) and both may say that it is black.
Let C = event that the ball drawn is black and both agreeing that it is black
P(C)=16×23×45=445
Let D = event that the ball drawn is non-black (any of the remaining 5 different colours) and both agreeing that it is black but telling a lie.
Both can tell a lie in 5 different ways.
Suppose the non- black ball drawn is blue, they may say it is black, white,red,yellow,green (all lies) of which probability of telling black is 15 on which they agree.
A may tell a lie that it is black in 15×13=115 ways.
B can tell it in 15×15=125 ways.
Also there is 56 probability of picking up a non-black ball.
Thus, probability that a non-black ball is drawn and both agree that it is black thus asserting that it is black but telling a lie is
P(D)=56×115×125=1450
Now, we have found probability of two events in which both are asserting that the ball is black, but when event C occurs the assertion is with speaking the truth and when event D occurs, assertion is by telling a lie.
The probability of their asserting that the ball is black and speaking truth
=P(C)P(C)+P(D)
=445445+1450
=4041