Coin Toss Probability Challenge

Solution & Explanation

The problem of calculating the expected number of coin flips required to achieve two consecutive heads is a classic example of a problem that can be solved using probability concepts, specifically the Negative Binomial Distribution.

Understanding the Problem

• We need to compute the expected number of coin tosses until we achieve two consecutive heads.
• A fair coin is used, meaning the probability of getting a head (H) in a single toss is $p = 0.5$.
• We are looking for two consecutive heads, denoted as HH.

Approach

1. Negative Binomial Distribution:

   • This distribution models the number of trials required to achieve a fixed number of successes in a sequence of independent Bernoulli trials.
   • For our problem, a "success" is getting a head (H), and we want two consecutive successes.

2. Mathematical Formulation:

   • Let $n$ be the number of consecutive heads (successes) we want, which is 2.

   • The probability of success $p$ for each trial (coin flip) is 0.5.

   • The expected number of trials (coin flips) to achieve $n$ consecutive successes is given by:

     $E[X] = \frac{n}{p} = \frac{2}{0.5} = 4$

Explanation

• Expected Value Calculation:

  • The expected number of flips to get the first head is $2$ (since $1/p = 2$).
  • After the first head, the expected number of additional flips to get another head is again $2$.
  • Therefore, on average, it takes $4$ flips to get two consecutive heads.

• Intuition:

  • The first head can occur at any time, and then we are essentially starting over to get another head. Hence, the problem can be broken down into smaller independent trials.

Conclusion

The expected number of coin flips required to achieve two consecutive heads is 4. This result is derived using the properties of the negative binomial distribution and understanding the independence of coin tosses in achieving consecutive successes.

By breaking down the problem into manageable parts and applying the negative binomial distribution, we can effectively determine the expected number of trials needed to achieve our desired outcome of two consecutive heads.