Question

If a seed is planted, it has a 90% chance of growing into a healthy plant. If 12 seeds are planted, what is the probability that exactly 2 don't grow

Answer 1

0.2301 = 23.01% probability that exactly 2 don't grow.

Explanation:

For each seed planted, there are only two possible outcomes. Either it grows into a healthy plant, or it does not. The probability of a seed growing into a healthy plant is independent of any other seed, which means that the binomial probability distribution is used to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

90% chance of growing into a healthy plant.

This means that
`p = 0.9`

12 seeds are planted

This means that
`n = 12`

What is the probability that exactly 2 don't grow?

So 12 - 2 = 10 grow, which is
`P(X = 10)`. Then

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 10) = C_(12,10).(0.9)^(10).(0.1)^(2) = 0.2301`

0.2301 = 23.01% probability that exactly 2 don't grow.