Probability Puzzles

Problem Statement

Three friends in Seattle each told you it’s rainy, and each person has a 1/3 probability of lying. What is the probability that Seattle is rainy? Assume the probability of rain on any given day in Seattle is 1/4.

Solution

We can use the Bayes theorem to find the required probability. For this case let's use the theorem in odds form
$${P(H_{1}|E) \over P(H_{2}|E)} = {P(H_{1}) \over P(H_{2})} * {P(E|H_{1}) \over P(E|H_{2})} $$ Let's define our hypotheses, priors and evidence:

$H_{1}$ - "Seattle is rainy"
$H_{2}$ - "Seattle is not rainy"
$P(H_{1}) = 1/4 $ - prior probability of the first hypothesis
$P(H_{2}) = 3/4 $ - prior probability of the second hypothesis
Evidence - all 3 friends told us it's rainy

Now we just need to calculate the probability of evidence under Hypothesis 1 and Hypothesis 2. In both cases we assume that friend's inputs are independent.

$P(E | H1) = (2/3)^3$ - if it's rainy, all 3 friends had to tell the truth to generate the evidence we have
$P(E | H2) = (1/3)^3$ - if it's not rainy, all 3 friends had to lie

Let's plug these numbers in the odds ratio:
$${P(H_{1}|E) \over P(H_{2}|E)} = {1/4 \over 3/4} * {(2/3)^2 \over (1/3)^3} = {8 \over 3} $$ Finally, we can convert odds to the required probability as follows:
$$\pmb{P(H_{1}|E) = {8 \over (8 + 3)} = {8 \over 11} = 0.73}$$ Not convinced? Try the simulation below!

Simulation

Note that in this simulation we need to model the situation when all 3 friends saying that the day is rainy. Only experiments where this criterion is satisfied will be used to calculate the final probability - this way we simulate the conditional probability we need.

show / hide simulation code


import numpy as np

nrounds = 1000
three_friends_said_rain_cases = 0
rainy_days = 0

while three_friends_said_rain_cases < nrounds:
    is_rainy_day = np.random.binomial(1, 1/4, 1)[0]

    friends_said_rain = 0
    if is_rainy_day:
        friends_said_rain += sum(np.random.binomial(1, 2/3, 3))
    else:
        friends_said_rain += sum(np.random.binomial(1, 1/3, 3))

    if friends_said_rain == 3:
        three_friends_said_rain_cases += 1
        rainy_days += is_rainy_day

simulated_probability = rainy_days / nrounds