Coin Bag Probability Challenge

Solution & Explanation

To solve this problem, we need to determine the expected number of heads observed when flipping 100 coins, selected randomly and with replacement, from a bag containing three types of coins:

• Unbiased coins: Show heads and tails equally.
• Coins with heads on both sides: Always show heads.
• Coins with tails on both sides: Never show heads.

Step-by-Step Solution:

1. Define Variables:

   • Let $x$ be the number of unbiased coins.
   • Let $y$ be the number of coins with heads on both sides.
   • Let $z$ be the number of coins with tails on both sides.
   • Let $N = x + y + z$ be the total number of coins in the bag.

2. Probability of Drawing Each Type of Coin:

   • Probability of drawing an unbiased coin ($F$): $\frac{x}{N}$
   • Probability of drawing a coin with heads on both sides ($H$): $\frac{y}{N}$
   • Probability of drawing a coin with tails on both sides ($T$): $\frac{z}{N}$

3. Expected Number of Heads from Each Coin Type:

   • Unbiased Coin: Each flip has a 50% chance of being heads, so expected heads per flip is $0.5$.
   • Coin with Heads on Both Sides: Always shows heads, so expected heads per flip is $1$.
   • Coin with Tails on Both Sides: Never shows heads, so expected heads per flip is $0$.

4. Calculate the Overall Expected Number of Heads:

   • The expected number of heads from a single flip can be calculated by summing up the expected heads from each type of coin:

     $E(H) = \left( \frac{x}{N} \times 0.5 \right) + \left( \frac{y}{N} \times 1 \right) + \left( \frac{z}{N} \times 0 \right)$

     Simplifying this, we get:

     $E(H) = \frac{0.5x + y}{N}$

5. Expected Number of Heads in 100 Flips:

   • Since each coin flip is an independent event, the expected number of heads in 100 flips is:

     $E(100H) = 100 \times \left( \frac{0.5x + y}{N} \right)$

6. Conclusion:

   • The expected number of heads observed when flipping 100 coins randomly with replacement is given by:

     $\frac{100(0.5x + y)}{x + y + z}$

This formula accounts for the probabilities of drawing each type of coin and their respective contributions to the expected number of heads in 100 flips.