There are 3 people – A, B and C. Probability that A speaks truth is 3/10, probability that B speaks truth is 3/7 and probability that C speaks truth is 5/6. For a particular question asked, at most 2 people speak truth. All people answer to a particular question asked. What is the probability that C will speak truth for a particular question asked?
A) 61/84
B) None
C) 81/93
D) 11/101
Given : Probability that A speaks truth is 3/10, probability that B speaks truth is 3/7 and probability that C speaks truth is 5/6. For a particular question asked, at most 2 people speak truth. All people answer to a particular question asked. B speaks truth only when A does not speak truth,
To Find : Probability that C does not speak truth on a question
A) 61/84
B) None
C) 81/93
D) 11/101
Solution:
Probability that A speaks truth is 3/10,
Probability that A does not speaks truth is 1 – 3/10 = 7/10
probability that B speaks truth is 3/7 and
Probability that B does not speaks truth is 1 – 3/7 = 4/7
probability that C speaks truth is 5/6
Probability that C does not speaks truth is 1 – 5/6 = 1/6
, at most 2 people speak truth.
Hence 0 speak truth or 1 speak truth or 2 speak truth
No one truth
0 speak truth = (7/10)(4/7)(1/6) = 28/420
A truth + B truth + C truth
1 speak truth = (3/10) (4/7)(1/6) + (7/10)(3/7)(1/6) + (7/10)(4/7)(5/6)
= (12 + 21 + 140)/420
= 173/420
B speaks truth only when A does not speak truth, hence A and B speak truth together not possible
A and B truth B and C truth A and C truth
2 people speak truth. = (3/10)(3/7)(1/6) + (7/10)(3/7)(5/6) + (3/10)(4/7)(5/6)
= (9+105 + 60)/420
= 174/420
at most 2 people speak truth. = (28 + 173 + 174)/420 when B speaks truth only when A does not speak truth,
= 375/420
C speak truth =( 140 + 105 + 60) /420 = 305/420
probability that C does not speak truth on a question =305/375
= 61/75
Hence none is the correct answer
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