Question

If a seed is planted, it has a 90% chance of growing into a healthy plant.

If 6 seeds are planted, what is the probability that exactly 2 don't grow?

Answer 1

Answer 2

`\displaystyle(19,683)/(200,000)\text{ or }\approx 9.84\%`

Explanation:

For each planted seed, there is a 90% chance that it grows into a healthy plant, which means that there is a
`100\%-90\%=10\%` chance it does not grow into a healthy plant.

Since we are planting 6 seeds, we want to choose 2 that do not grow and 4 that do grow:

`\displaystyle (1)/(10)\cdot (1)/(10)\cdot (9)/(10)\cdot (9)/(10)\cdot (9)/(10)\cdot (9)/(10)`

However, this is only one case that meets the conditions. We can choose any 2 out of the 6 seeds to be the ones that don't grow into a healthy plant, not just the first and second ones. Therefore, we need to multiply this by number of ways we can choose 2 things from 6 (6 choose 2):

`\displaystyle \binom{6}{2}=(6\cdot 5)/(2!)=(30)/(2)=15`

Therefore, we have:

`\displaystyle\\P(\text{exactly 2 don't grow})=(1)/(10)\cdot (1)/(10)\cdot (9)/(10)\cdot (9)/(10)\cdot (9)/(10)\cdot (9)/(10)\cdot \binom{6}{2},\\\\P(\text{exactly 2 don't grow})=(1)/(10)\cdot (1)/(10)\cdot (9)/(10)\cdot (9)/(10)\cdot (9)/(10)\cdot (9)/(10)\cdot 15,\\\\P(\text{exactly 2 don't grow})=\boxed{(19,683)/(200,000)}\approx 9.84\%`