Question

You're about to get on a plane to Seattle. You want to know if you should bring an umbrella. You call 3 random friends of yours who live there and ask each independently if it's raining. Each of your friends has a 2/3 chance of telling you the truth and a 1/3 chance of messing with you by lying. All 3 friends tell you that "Yes" it is raining. What is the probability that it's actually raining in Seattle?

Answer 1

We will start by considering the various events:

• Probability of rain in Seattle is
• Probability of not raining in Seattle is
• Probability of all three friends saying yes is
• Probability of all three friends saying yes given it is actually raining is
• Probability of all three friends saying yes given it is actually not raining is
• Probability that it's actually raining given all three friends said yes is ​. This is the value we need to evaluate in our question.

The first observation is that when all three friends say it is raining, it is either that all three are speaking truth or all three are lying. If it is actually raining, all three are speaking truth, else all three are lying.

Given that ​ and ​ . Thus ​ and ​

Now probability of all three saying yes ​is given by,

The formula for Baye's theorem is given by,

Using baye's formula, the probability that it's actually raining given all three friends said yes​ is given by,

  

The probability that it's actually raining given all three friends said yes​ is

SPECIFIC SCENARIOS:

1) Assume that the prior probabilities of raining :

Then probability of all three saying yes ​is given by,

The probability that it's actually raining given all three friends said yes​ is given by,

2) Assume that the prior probabilities of raining :

Then probability of all three saying yes ​is given by,

The probability that it's actually raining given all three friends said yes​ is given by,