Chance of Even Heads

Solution & Explanation

To solve the problem of determining the probability of obtaining an even number of heads when flipping a fair coin 1,000 times, we can utilize concepts from probability theory and binomial distribution.

1. Understanding the Binomial Distribution

A fair coin flip is a classic example of a Bernoulli trial, where there are two possible outcomes: heads or tails. When conducting multiple independent Bernoulli trials, we can model the situation using a binomial distribution.

• Number of Trials (n): 1000
• Probability of Success (p): 0.5 (since the coin is fair)

The probability of obtaining exactly k heads in n flips is given by the binomial probability formula:

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

2. Even vs. Odd Number of Heads

We are interested in the probability of achieving an even number of heads. This means we need to find the sum of probabilities for all even k values from 0 to 1000:

$P(\text{even heads}) = \sum_{k=0}^{500} \binom{1000}{2k} (0.5)^{1000}$

3. Symmetry of the Binomial Distribution

The key insight here is the symmetry of the binomial distribution when p = 0.5. Given the symmetry:

• The probability of getting an even number of heads is equal to the probability of getting an odd number of heads.

Since there are only two possibilities (even or odd), and they are equally likely due to the symmetry:

$P(\text{even heads}) = P(\text{odd heads}) = 0.5$

4. Intuitive Explanation Using the Last Flip

Another way to understand this is to consider the role of the last coin flip:

• If the first 999 flips result in an even number of heads, the last flip must be a tail to maintain an even count, and vice versa.
• Thus, for any sequence of 999 flips, the last flip will determine the parity of the total number of heads.

Given the independence of each flip, the probability that the last flip will make the total even is 0.5.

Conclusion

Whether approached through the binomial distribution or intuitive reasoning, the probability of getting an even number of heads in 1,000 flips of a fair coin is 0.5. This reflects the inherent symmetry and balance of the binomial distribution when dealing with a fair coin. This problem illustrates the elegance and simplicity of probability theory in capturing the essence of random processes.