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Question
Often it is taken that a truthful person commands, more respect in the society. A man is known to speak the truth 4 out of 5 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.
Do you also agree that the value of truthfulness leads to more respect in the society?
Solution
Let `E_1` : The event that six comes on the die.
`E_2`: The event that six does not comes on the die.
A : The event that man reports it is a six.
`:. P(E_1) = 1/6` and `P(E_2) = 1 - P(E_1)`
`= 1 - 1/6`
`= 5/6`
Probability that the man report that there is a six on the die given that six comes on the die
`= P(A/E_1)`
= Probability that man speaks truth
= `4/5`
Probability that the man reports that there is a six on the die given that six does not comes on the die
= `P(A/E_2)`
= Probability that man does not speak truth
`= 1 - 4/5`
`= 1/5`
By Baye's Theorem, we have
`P(E_1/A)` = Probability that there is a six given that man reports that there is a six on die.
`= (P(E_1)P(A/E_1))/(P(E_1)P(A/E_1) + P(E_2)P(A/E_2))`
`= (1/6 xx 4/5)/(1/6 xx 4/5 + 5/6 xx 1/5)`
`= 4/9`
Yes, truthfulness always lead to respect in the society as truth always wins.
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