Financial Risk Manager Part 1

Explanation

This is a classic Bayes' theorem problem. Let's break it down step by step:

Given:

• Coin 1: Double-headed (always lands on heads)
• Coin 2: Normal unbiased coin (50% chance of heads)
• Both coins are equally likely to be selected: P(coin 1) = P(coin 2) = 1/2
• We observe a head after the toss

Applying Bayes' Theorem:

We want to find P(coin 1 | head)

$\text{P(coin 1 | head)} = \frac{\text{P(coin 1)} \times \text{P(head | coin 1)}}{\text{P(coin 1)} \times \text{P(head | coin 1)} + \text{P(coin 2)} \times \text{P(head | coin 2)}}$

Calculating probabilities:

• P(coin 1) = 1/2
• P(head | coin 1) = 1 (since coin 1 is double-headed)
• P(coin 2) = 1/2
• P(head | coin 2) = 1/2 (since coin 2 is unbiased)

Substituting values:

$\text{P(coin 1 | head)} = \frac{(\frac{1}{2} \times 1)}{(\frac{1}{2} \times 1) + (\frac{1}{2} \times \frac{1}{2})} = \frac{\frac{1}{2}}{\frac{1}{2} + \frac{1}{4}} = \frac{\frac{1}{2}}{\frac{3}{4}} = \frac{1}{2} \times \frac{4}{3} = \frac{2}{3}$

Intuitive understanding:

When we see a head, it could come from either coin. However, coin 1 always produces heads, while coin 2 only produces heads half the time. Therefore, when we observe a head, it's more likely to have come from the double-headed coin than from the fair coin.

The probability $\frac{2}{3}$ means that given we observed a head, there's a 2/3 chance it came from the double-headed coin and a 1/3 chance it came from the fair coin.