If a seed is planted, it has a 60% chance of growing into a healthy plant.

If 6 seeds are planted, what is the probability that exactly 4 don't grow?

To find the probability that exactly 4 seeds don't grow, we need to calculate the probability for each possible combination of 4 seeds not growing.

The probability that a seed doesn't grow is 1 - 0.6 = 0.4. So, the probability that a seed does grow is 0.6.

To calculate the probability of exactly 4 out of 6 seeds not growing, we can use the binomial distribution formula:

P(X=k) = (nCk) * p^k * (1-p)^(n-k)

Where:
- P(X=k) is the probability of k successes (seeds not growing)
- nCk is the combination of n items taken k at a time
- p is the probability of a success (seed not growing)
- n is the total number of trials (seeds planted)
- k is the number of successes (seeds not growing)

In this case, n = 6, p = 0.4, and k = 4. Let's calculate:

P(X=4) = (6C4) * 0.4^4 * (1-0.4)^(6-4)

= (6C4) * 0.4^4 * 0.6^2

To calculate (6C4), we can use the combination formula:

(6C4) = 6! / (4! * (6-4)!)

= 6! / (4! * 2!)

= (6*5*4*3*2*1) / ((4*3*2*1) * (2*1))

= 15

Now, let's substitute the values into the main formula:

P(X=4) = (6C4) * 0.4^4 * 0.6^2

= 15 * 0.4^4 * 0.6^2

= 15 * 0.0256 * 0.36

= 0.1377

Therefore, the probability that exactly 4 out of 6 seeds don't grow is approximately 0.1377, or 13.77%.

To calculate the probability, we need to consider the total number of possible outcomes and the number of favorable outcomes.

Let's break down the problem:

1. Number of Seeds Planted: 6
2. Probability of Not Growing: 40% or 0.4 (since there is a 60% chance of growing, the probability of not growing is the complement of that, which is 1 - 0.6 = 0.4)
3. Number of Seeds That Don't Grow: 4

To find the probability that exactly 4 seeds don't grow, we need to calculate the probability of this specific combination happening. We can use the binomial probability formula:

P(X=k) = nCk * p^k * (1-p)^(n-k)

Where:
- P(X=k) is the probability of getting exactly k successes
- nCk is the number of combinations of n items taken k at a time
- p is the probability of success
- (1-p) is the probability of failure
- n is the total number of trials
- k is the number of desired successes

In this case, n = 6 (the total number of seeds planted), k = 4 (the number of seeds that don't grow), and p = 0.4 (the probability of not growing a seed).

Using the formula, we can calculate the probability as follows:

P(X=4) = 6C4 * (0.4)^4 * (1-0.4)^(6-4)

Now, let's calculate step by step:

1. Number of Combinations (6C4):
- 6C4 = 6! / (4! * (6-4)!) = 6! / (4! * 2!) = (6 * 5 * 4!) / (4! * 2 * 1) = (6 * 5) / (2 * 1) = 15

2. Probability of 4 Seeds Not Growing:
- (0.4)^4 = 0.4 * 0.4 * 0.4 * 0.4 = 0.0256

3. Probability of 2 Seeds Growing:
- (1 - 0.4)^(6 - 4) = 0.6^2 = 0.36

4. Final Calculation:
- P(X=4) = 15 * 0.0256 * 0.36 ≈ 0.13824

Therefore, the probability that exactly 4 out of 6 seeds don't grow is approximately 0.13824 or 13.824%.

Here, let's define "success" to be that the plant doesn't grow. This would be 1 - .6 = .4.

We construct a binomial distributions with 6 trials (6 seeds planted). We want 4 events.

B(6, .4)
P(X = 4) = .138

6C4 .4^4 .6^2 = 0.138