easy | probability |

A father claims about snowfall last night. First daughter tells that the probability of snowfall on a particular night is 1/8. Second daughter tells that 5 out of 6 times the father is lying! What is the probability that there actually was a snowfall?

Conditional Probability or Baye's Theorem

1/36

Let $S =$ Snowfall occurred, and $C$ be the event that the father is claiming snowfall ocurred.

Probability of (Snowfall given Claim) = $P(S | C) = \dfrac{P(C|S) \cdot P(S)}{P(C)}$

$P(S) = 1/8$

$P(C|S) =1 - 5/6 = 1/6$ (given in the question, father lies 5/6 times)

$P(C) = P(\text{True claim}) + P(\text{False Claim})$

$= P(C \cap S) + P(C \cap S')$ where $S'$ means that it did not snow.

$= P(C|S) \cdot P(S) + P( C | S') * P(S')$

$= (1/6 \cdot 1/8) + (5/6 \cdot 7/8)$

$P(S | C) = \dfrac{P(C|S) \cdot P(S)}{P(C)} = \dfrac{1/6 \cdot 1/8}{(1/6 \cdot 1/8) + (5/6 \cdot 7/8)} = \dfrac{1}{36}$