Question

If a seed is planted, it has a 70% chance of growing into a healthy plant.

If 6 seeds are planted, what is the probability that exactly 3 don't grow?

Answer 1

To find the probability that exactly 3 out of 6 seeds do not grow, we can use the binomial probability formula. In this scenario, the number of seeds that grow follows a binomial distribution.

Let:
- \( n = 6 \) (the number of seeds planted),
- \( p = 0.7 \) (the probability that a seed grows),
- \( q = 1 - p = 0.3 \) (the probability that a seed does not grow),
- \( k = 3 \) (the number of seeds that do not grow).

We want to find the probability that exactly 3 seeds do not grow, which means that 3 seeds do grow. The binomial probability formula is given by:

\[
P(X = k) = \binom{n}{k} p^{n-k} q^k
\]

In this case, since we want the probability that exactly 3 do not grow, we can express it as:

\[
P(\text{exactly 3 do not grow}) = P(X = 3) = \binom{6}{3} (0.7)^{6-3} (0.3)^3
\]

Calculating step-by-step:

1. Calculate \( \binom{6}{3} \):
\[
\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20
\]

2. Calculate \( (0.7)^{3} \):
\[
(0.7)^3 = 0.343
\]

3. Calculate \( (0.3)^{3} \):
\[
(0.3)^3 = 0.027
\]

4. Now combine all parts to find \( P(X = 3) \):
\[
P(X = 3) = 20 \times 0.343 \times 0.027
\]
\[
= 20 \times 0.009261 = 0.18522
\]

Thus, the probability that exactly 3 out of the 6 seeds do not grow is approximately **0.185** or **18.5%**.