Question

A box contains 5 coins: 4 regular coins and 1 fake coin.

- When a regular coin is tossed, the probability $P(\text{head}) = 0.5$.
- For a fake coin, $P(\text{head}) = 1$.

You pick a coin at random and toss it twice, and get two heads. The probability that the coin you have chosen is the fake coin is ______.

Hint:

Bayes' Theorem allows us to update the probability of an event based on new evidence. In this case, it helps us find the probability of having chosen the fake coin, given the outcome of two heads.

Answer 1

We are asked to find the probability that the coin is fake, given that two heads were tossed.

This is a problem of conditional probability. We can use Bayes' Theorem to solve it. Bayes' Theorem is given by:

$$P(\text{Fake} \mid \text{Two heads}) = \frac{P(\text{Two heads} \mid \text{Fake}) P(\text{Fake})}{P(\text{Two heads})}$$ We will calculate each term in this formula:

1. Prior probability of choosing the fake coin:
Since there is 1 fake coin out of 5 coins, the probability of selecting the fake coin is:
$$P(\text{Fake}) = \frac{1}{5}$$
2. Likelihood of getting two heads given the fake coin:
If the fake coin is chosen, the probability of getting heads on each toss is 1. Therefore, the probability of getting two heads is:
$$P(\text{Two heads} \mid \text{Fake}) = 1 \times 1 = 1$$
3. Likelihood of getting two heads given a regular coin:
If a regular coin is chosen, the probability of getting heads on each toss is 0.5. Therefore, the probability of getting two heads is:
$$P(\text{Two heads} \mid \text{Regular}) = 0.5 \times 0.5 = 0.25$$
4. Prior probability of choosing a regular coin:
There are 4 regular coins out of 5, so the probability of selecting a regular coin is:
$$P(\text{Regular}) = \frac{4}{5}$$
5. Total probability of getting two heads:
The total probability of getting two heads is given by the law of total probability:
$$P(\text{Two heads}) = P(\text{Two heads} \mid \text{Fake}) P(\text{Fake}) + P(\text{Two heads} \mid \text{Regular}) P(\text{Regular})$$ Substituting the values:
$$P(\text{Two heads}) = 1 \times \frac{1}{5} + 0.25 \times \frac{4}{5} = \frac{1}{5} + \frac{1}{5} = \frac{2}{5}$$
6. Bayes' Theorem:
Now, applying Bayes' Theorem:
$$P(\text{Fake} \mid \text{Two heads}) = \frac{P(\text{Two heads} \mid \text{Fake}) P(\text{Fake})}{P(\text{Two heads})} = \frac{1 \times \frac{1}{5}}{\frac{2}{5}} = \frac{1}{2}$$
Thus, the probability that the coin you have chosen is the fake coin is: $\boxed{0.50}$.