Comment on Activity 1

Table E1 shows the possible outcomes for three tosses.

Table E1   Outcomes of Three Flips of a Fair Coin
 
Toss 1
Toss 2
Toss 3
Outcome 1
H
H
H
Outcome 2
H
H
T
Outcome 3
H
T
H
Outcome 4
H
T
T
Outcome 5
T
H
H
Outcome 6
T
H
T
Outcome 7
T
T
H
Outcome 8
T
T
T

The probability of at least one head is `7//8`.

In general, the probability of at least one head in `n` tosses is (the probability of a head on the first toss) plus (the probability of a tail on the first toss and a head on the second toss) plus `...` plus (the probability of tails on the first `n-1` tosses and a head on the `n`th toss). In symbols this is

`1/2+1/(2^2)+ cdots + 1/(2^k)+ cdots + 1/(2^n)`.

This is the same as `1` minus the probability of tails on all tosses. Since only one of the `2^n` outcomes has all tails, the probability of at least one head in `n` tosses is

`1-1/2^n = (2^n - 1)/2^n`.

In particular, for `n=5`, the probability of at least one head is `text[(]2^5 - 1text[)/]2^5 = 31text[/]32 = 0.96875`.