What is the probability of a biased coin coming up heads given that a liar is claiming that the coin came up heads?

Solution 1:

You're correct, an easy way to check this is by writing out the possible outcomes. In a perfect world, if we flip the coin 32 times the following will happen:

• 21 times it lands tails and the liar said head.
• 7 times it lands tails and the liar said tails.
• 3 times it lands head and the liar said tails.
• 1 time it lands head and the liar said head.

Since it is given that the liar said head there are 22 options left over, only one of which has the coin actually landing head.

I know this is not the proper way of solving this, but I always found it useful to write things out like this when I was getting confused about conditional probability.

Solution 2:

The Question: Is the method I used wrong in any way?

Nope, your solution is accurate. The coin is rarely heads and truthfully said to be so. The coin is much more often tails yet said to be heads. So when the result is said to be heads, the coin is quite unlikely to truly be heads.$$\tfrac{\tfrac 18\tfrac 14}{\tfrac 18\tfrac 14+\tfrac 78\tfrac 34}=\dfrac 1{22}$$

Some others I have talked to are saying the answer will be $1/4$. Their reasoning is this: since the liar lies $3$ times out of $4$ and he said it is head, then the probability of it being head is $1/4$. So who is right? What will be the answer?

Consider using another coin, one with two tails - so the result cannot truly be heads - while the reporter lies with the same probability as above. So if this coin is flipped and said to be heads, what is the (conditional) probability that the result is truly heads?

Your method says: $0$, while their method says $1/4$.

Solution 3:

I find a nice way to think about the problem is a table. First let's declare probabilities

P(Heads) = 1/8
P(Tails) = 7/8
P(Lie)   = 3/4
P(Truth) = 1/4

With this information we can make the following table

+-------+------------+------------+
|       | Says Heads | Says Tails |
+-------+------------+------------+
| Heads | (1/8)(1/4) | (1/8)(3/4) |
+-------+------------+------------+
| Tails | (7/8)(3/4) | (7/8)(1/4) |
+-------+------------+------------+ 

Since the liar already told us heads we can ignore the right column. So the Probability of $P(Heads)P(Truth)$ would be $P(Heads)P(Truth)$ over the sum of all other possibilities where the liar says heads, or

$$ \dfrac{ P(Heads)P(Truth) }{ P(Heads)P(Truth) + P(Tails)P(Lie) } = 1/22 $$

Your approach is correct but this is another way to justify the number you arrive to.