Question

Let \(X\) represent the difference between the number of heads and the number of tails obtained when a coin is tossed \(n\) times. What are the possible values of \(X\) ?

Short answer

The possible values for \(X\) are the integers in the range \(-n \leq X \leq n\), in steps of 1.

Step-by-step solution

Analyze and define the variables

When a coin is tossed \(n\) times, there can be anywhere from 0 heads to \(n\) heads as a result, with the remaining coin flips resulting in tails. Let's denote the number of heads as variable \(H\) and the number of tails as variable \(T\). The difference between heads and tails is represented by \(X = H - T\).

Understand the minimum and maximum values of X

The minimum value for \(H\) is 0, meaning all coin flips result in a tail. The maximum value for \(H\) is \(n\), meaning all coin flips result in a head. Conversely, the minimum value for \(T\) is also 0 and the maximum is \(n\). To get the minimum value for \(X\), we'll have \(H=0\) and \(T=n\). Plugging these values, we get \(X_{min} = 0 - n = -n\). To get the maximum value for \(X\), we'll have \(H=n\) and \(T=0\). Plugging these values, we get \(X_{max} =n-0 = n\).

Determine the possible values for X

Now, we have established that the minimum value for \(X\) is \(-n\) and the maximum value is \(n\). Since \(H\) and \(T\) can only be integers, \(X\) will also be an integer. Therefore, the possible values of \(X\) range from \(-n\) to \(n\) in integer steps. In conclusion, the possible values for \(X\) are the integers in the range \(-n \leq X \leq n\), in steps of 1.