Question

A coin is tossed 16 times. What is the probability of obtaining exactly 14 heads?

Answer 1

`P("14 heads in 16 tosses of a fair coin")=120/65536~=0.18%`

Explanation:

When calculating a probability, we take the ratio of the number of ways to meet a certain condition (i.e. the numerator) divided by the number of ways to pick from a pool (i.e. the denominator).

So what are the number of ways the flip of a coin 16 times can come out? Each toss has two possible results - if we toss twice we have 4 `(=2^2)` possible results, thrice we have 8 `(=2^3)` possible results, etc. It works out to be, for 16 tosses:

`2^16=65,536`

Of those results, how many ways can we achieve 14 heads in 16 tosses?

This is a combinations calculation, with the general formula:

`C_(n,k)=(n!)/((k!)(n-k)!)` with `n="population", k="picks"`

`C_(16,14)=(16!)/((14!)(16-14)!)=(16!)/(2!xx14!)=(16xx15xx14!)/(2xx14!)=120`

And so we have:

`P("14 heads in 16 tosses of a fair coin")=120/65536~=0.18%`