Biased Coin Probability

Solution & Explanation

The problem involves determining the probability that you selected the biased coin after observing 10 heads in a row from your chosen coin. This is a classic example of using Bayes' Theorem to calculate conditional probabilities.

Bayes' Theorem

Bayes' Theorem is a mathematical formula used to determine the conditional probability of an event, given the probability of another event that has already occurred. The theorem is expressed as:

$P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}$

Where:

• $P(A|B)$ is the probability that event $A$ occurs given that $B$ is true.
• $P(B|A)$ is the probability that event $B$ occurs given that $A$ is true.
• $P(A)$ is the probability of event $A$ occurring.
• $P(B)$ is the probability of event $B$ occurring.

Definitions for this Problem

• Event $A$: The coin selected is the biased coin (two heads).
• Event $A'$: The coin selected is a fair coin (one head, one tail).
• Event $B$: The result of flipping the coin 10 times is 10 heads.

Given Probabilities

• $P(A) = \frac{1}{100}$ because there is 1 biased coin out of 100 coins.
• $P(A') = \frac{99}{100}$ because there are 99 fair coins.
• $P(B|A) = 1$ because the biased coin will always show heads.
• $P(B|A') = \left(\frac{1}{2}\right)^{10}$ because a fair coin has a $\frac{1}{2}$ chance of showing heads per flip.

Calculate $P(B)$

To find $P(B)$, use the law of total probability:

$P(B) = P(B|A) \times P(A) + P(B|A') \times P(A')$

Substitute the values:

$P(B) = (1 \times \frac{1}{100}) + \left(\left(\frac{1}{2}\right)^{10} \times \frac{99}{100}\right)$

Simplifying further:

$P(B) = \frac{1}{100} + \frac{99}{1024 \times 100}$

Calculate $P(A|B)$

Now apply Bayes' Theorem:

$P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}$

Substitute the known values:

$P(A|B) = \frac{1 \times \frac{1}{100}}{\frac{1}{100} + \frac{99}{1024 \times 100}}$

Simplify the expression:

$P(A|B) = \frac{1}{1 + \frac{99}{1024}} = \frac{1024}{1123} \approx 0.9118$

Conclusion

The probability that the coin you selected is the biased one, given that you observed 10 heads in a row, is approximately 0.9118. This high probability reflects how unlikely it is to get 10 consecutive heads with a fair coin, thus making it more probable that the biased coin was chosen.