Understanding the Coin Toss Problem
When a coin is tossed \( n \) times, it can result in a combination of heads and tails. The goal is to find the probability that the difference between the number of heads and tails is \( n - 3 \).
Difference Calculation
- Let \( H \) represent the number of heads and \( T \) represent the number of tails.
- The difference can be expressed as:
\[
|H - T| = n - 3
\]
- Since \( H + T = n \), we can derive two scenarios:
1. \( H - T = n - 3 \)
2. \( T - H = n - 3 \)
Solving the Equations
- From \( H - T = n - 3 \):
\[
H + (H - (n - 3)) = n \implies 2H - n + 3 = n \implies 2H = 2n - 3 \implies H = n - \frac{3}{2}
\]
- From \( T - H = n - 3 \):
\[
T + (T - (n - 3)) = n \implies 2T - n + 3 = n \implies 2T = 2n - 3 \implies T = n - \frac{3}{2}
\]
Conditions for Validity
- \( H \) and \( T \) must be non-negative integers. Thus, we need \( n \) to be an odd number (to ensure \( \frac{3}{2} \) results in an integer).
Probability Calculation
- The total number of outcomes when tossing a coin \( n \) times is \( 2^n \).
- The number of favorable outcomes for either scenario (heads or tails) can be calculated using the binomial coefficient.
- The final probability is given by:
\[
P = \frac{\text{Number of favorable outcomes}}{2^n}
\]
In summary, the probability that the difference between the number of heads and tails is \( n - 3 \) will depend on the appropriate combinations and whether \( n \) is odd. For precise calculations, use the binomial distribution.