Probability that A speaks truth is \(\frac{4}{5}\).A coin is tossed.A report that a head appears.The probability that actually there was head is:
Probability that A speaks truth is \(\frac{4}{5}\).A coin is tossed.A report that a head appears.The probability that actually there was head is:
(\(\frac{4}{5}\))
(\(\frac{1}{2}\))
(\(\frac{1}{5}\))
(\(\frac{2}{5}\))
The Correct Option is A
Solution and Explanation
Let A be the event that the man reports that head occurs in tossing a coin and let E1 be the event that head occurs and E2 be the event head does not occur.
P(E1)=\(\frac{1}{2}\),P(E2)=\(\frac{1}{2}\)
P(A|E1)=P(A reports that head occurs when head had actually occur red on the coin)=\(\frac{4}{5}\)
P(A|E2)=P(A reports that leads occurs when head had not occur red on the coin)=1-\(\frac{4}{5}\)=\(\frac{1}{5}\)
By Bayes'theorem,
P(E1|A)=\(\frac{P(E1)P(A|E_1)}{P(E_1)P(A|E_1)+P(E_2)P(A|E_2)}\)=\(\frac{\frac{1}{2}×\frac{4}{5}}{\frac{1}{2}×\frac{4}{5}+\frac{1}{2}×\frac{1}{5}}\)=4/4+1=4/5
Hence,option (A) is correct.
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Concepts Used:
Bayes Theorem
Bayes’ Theorem is a part of the conditional probability that helps in finding the probability of an event, based on previous knowledge of conditions that might be related to that event.
Mathematically, Bayes’ Theorem is stated as:-
\(P(A|B)=\frac{P(B|A)P(A)}{P(B)}\)
where,
- Events A and B are mutually exhaustive events.
- P(A) and P(B) are the probabilities of events A and B, respectively.
- P(A|B) is the conditional probability of the happening of event A, given that event B has happened.
- P(B|A) is the conditional probability of the happening of event B, given that event A has already happened.
This formula confines well as long as there are only two events. However, Bayes’ Theorem is not confined to two events. Hence, for more events.