Question

3-a coin is tossed three times, what is the probability of tossing exactly two heads?

Answer 1

`3/8`

Explanation:

Since 3 is a small number, let's list out all possible combinations.

`color(green){H}` represents a head while `color(red){T}` represents a tail.

`color(blue){3" Heads"}`

`color(green){HHH}`

`color(blue){2" Heads"}`

`color(green){HH}color(red){T}`
`color(green){H}color(red){T}color(green){H}`
`color(red){T}color(green){HH}`

`color(blue){1" Head"}`

`color(green){H}color(red){T T}`
`color(red){T}color(green){H}color(red){T}`
`color(red){T T}color(green){H}`

`color(blue){0" Head"}`

`color(red){T T T}`

The answer is `frac{3}{1+3+3+1} = 3/8`.

In general, you will find that the list resembles a particular row of the pascal's triangle.

If you want to know what is the probability of getting `r` heads (or tails) out of `n` flips, it is the `r^{"th"}` element in the `n+1` row, divided by the sum of all the elements in the `n+1` row. Mathematically, the expression is

`frac{((n),(r))}{2^n} = frac{n!}{r! xx (n-r)! xx 2^n}`

In this case, `n = 3` and `r = 2`.

So

`frac{3!}{2! xx 1! xx 2^3} = frac{6}{2 xx 1 xx 8}`

`= 3/8`