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A speaks the truth 2 out of 3 times and B 4 out of 5 times. They agree in the assertion that from a bag containing 6 balls of different colours a black ball has been drawn. Find the probability that the statements are true?

A speaks the truth 2 out of 3 times and B 4 out of 5 times. They agree in the assertion that from a bag containing 6 balls of different colours a black ball has been drawn. Find the probability that the statements are true?

Grade:12

1 Answers

Arun
25750 Points
3 years ago
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)= 
6
1
​ 
 × 
3
2
​ 
 × 
5
4
​ 
 = 
45
4
​ 
 
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  
5
1
​ 
  on which they agree.
A may tell a lie that it is black in  
5
1
​ 
 × 
3
1
​ 
 = 
15
1
​ 
  ways.
B can tell it in  
5
1
​ 
 × 
5
1
​ 
 = 
25
1
​ 
  ways.
Also there is  
6
5
​ 
  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)= 
6
5
​ 
 × 
15
1
​ 
 × 
25
1
​ 
 = 
450
1
​ 
 
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(D)
P(C)
​ 
 
= 
45
4
​ 
 + 
450
1
​ 
 
45
4
​ 
 
​ 
 
= 
41
40
​ 

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