Correlation in Coin Toss Experiments

The question asks for the correlation coefficient between the number of heads and the number of tails in an experiment consisting of tossing a coin 20 times. Let the number of heads in a single experiment be $\text{H}$ and the number of tails be $\text{T}$.

Relationship Between Heads and Tails

When a coin is tossed 20 times, the total number of outcomes is 20. This total is the sum of the number of heads and the number of tails. Therefore, for any single experiment:

$\text{H} + \text{T} = 20$

This equation shows a direct relationship between $\text{H}$ and $\text{T}$. We can rearrange this equation to express $\text{T}$ in terms of $\text{H}$:

$\text{T} = 20 - \text{H}$

This equation represents a linear relationship between $\text{H}$ and $\text{T}$. The slope of this linear relationship is -1 (for every increase of 1 in $\text{H}$, $\text{T}$ decreases by 1). This is a perfect inverse relationship.

Understanding Correlation Coefficient

The correlation coefficient is a statistical measure that quantifies the strength and direction of a linear relationship between two variables. Its value ranges from -1 to +1:

• A value of +1 indicates a perfect positive linear relationship (as one variable increases, the other increases proportionally).
• A value of -1 indicates a perfect negative linear relationship (as one variable increases, the other decreases proportionally).
• A value of 0 indicates no linear relationship.

Correlation Between Heads and Tails

Since the relationship between the number of heads ($\text{H}$) and the number of tails ($\text{T}$) in a 20-toss experiment is $\text{T} = 20 - \text{H}$, this is a perfect negative linear relationship. Regardless of the specific outcomes of the tosses (within the 20 tosses), if you know the number of heads, you can perfectly determine the number of tails, and vice versa. As the number of heads increases, the number of tails decreases by the exact same amount to maintain the sum of 20.

This perfect negative linear association means that the correlation coefficient between the number of heads and the number of tails is -1.

The fact that the experiment is performed 50 times does not change the fundamental relationship between $\text{H}$ and $\text{T}$ within each individual experiment of 20 tosses. The correlation coefficient describes this specific relationship, not something between the 50 repetitions.

Conclusion

Based on the perfect negative linear relationship $\text{T} = 20 - \text{H}$, the correlation coefficient between the number of heads and the number of tails in a 20-toss experiment is -1.