4

Karan tells truth with probability $\frac 13$ and lies with probability $\frac 23$. Independently, Arjun tells truth with probability $\frac 34$ and lies with probability $\frac 14$. Both watch a cricket match. Arjun tells you that India won, Karan tells you that India lost. What probability will you assign to India's win?

$(a) \frac 12$

$(b)\frac 23$

$(c)\frac 34$

$(d)\frac 56$

$(e)\frac 67$

According to me the answer should be $\frac 12$ i.e option $(a)$

Arjun told: India won,

Karan told: India lost,

Probability of India won = Probability that Arjun told truth$(=\frac 34)$ & Karan lied$(= \frac 23)$

So probability that India won = $\frac 34\times\frac 23 =\frac 12$. Is it correct?

Romy
  • 279
  • According to your calculations, the probability that India loses will be P(Arjun lies)P(Karan tells the truth) = 1/4 1/3 = 1/12 $\neq$ 1 - 1/2 –  Oct 05 '15 at 13:27
  • Another way to see that your method can't be correct: suppose that Arjun always tells the truth. Then we know that India in fact did win, but your method would just give a probability of $\frac 34$. – lulu Oct 05 '15 at 13:28
  • (Win) + P(Loss) need not be equal to 1, since there can be logical inconsistencies on the reports I got. In this case for example: There are 4 possible reports:
    1. Arjun Told Truth & Karan Told Truth,
    2. Arjun Told Truth & Karan Lied,
    3. Arjun Lied & Karan Told Truth,
    4. Both of them lied.

    out of these 4 reports 2 are logically inconsistent in which both of them told truth or both of them lied. So It should be P(Win) + P(Loss) + P(Inconsistencies) = 1.

     – Romy Oct 05 '15 at 13:30
  • You need to use Bayes rule. – A.S. Oct 05 '15 at 13:32

1 Answers1

3

Suppose that India won. In that case the probability that Arjun said that India won is $\frac34$, and the probability that Karan lied is $\frac23$, so if India won, the probability of the observed statements is $\frac34\cdot\frac23=\frac12$.

Now suppose that India lost, so that Arjun is lying, and Karan is telling the truth. Thus, if India lost, the probability of the observed statements is $\frac14\cdot\frac13=\frac1{12}$.

As you can see, we’re much likelier to get the observed statements if India won than we are if India lost. Thus, the probability that India won must surely be more than $\frac12$. In fact Bayes’ theorem says that the probability that India won is the fraction of the total probability $\frac12+\frac1{12}$ that is contributed by the first case, i.e.,

$$\frac{\frac12}{\frac12+\frac1{12}}=\frac67\;.$$

Brian M. Scott
  • 616,228