How is the probability of getting at most one head in several coin tosses calculated?

To calculate the probability of getting at most one head in $n$ tosses of a fair coin, you must determine the ratio of favorable outcomes to the total number of possible outcomes in the sample space.

Since each of the $n$ tosses has two possible results (heads or tails), the total number of elements in the sample space is $2^n$. To find the number of favorable cases for "at most one head," you sum the outcomes for two mutually exclusive scenarios:

• Zero heads: There is only $1$ outcome where no heads occur (all tails).
• Exactly one head: There are $n$ possible outcomes where exactly one head occurs, as the head can appear on the first, second, ..., or $n$th toss.

By adding these together, you find there are $n+1$ favorable cases. Therefore, the probability of obtaining at most one head is:

$$\frac{n+1}{2^n}$$

This calculation relies on the assumption that all outcomes in the sample space are equally likely.1