If a seed is planted, it has a 65% chance of growing into a healthy plant.
If 11 seeds are planted, what is the probability that exactly 4 don't grow?
this is a binary probability ... Grow or Not grow
... p(G) = .65 ... p(N) = .35
(G + N)^11 = 11C11 G^11 + 11C10 G^10 N + ... + 11C0 N^11
you want the 5th term ... 11C7 G^7 N^4 ... 11C7 * .65^7 * .35^4
To find the probability that exactly 4 out of 11 seeds don't grow, we can use the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
where:
- P(X = k) is the probability of getting exactly k successful outcomes,
- C(n, k) is the number of ways to choose k items from a set of n (also known as the binomial coefficient),
- p is the probability of success for each individual trial (in this case, the probability that a seed doesn't grow), and
- n is the total number of trials (in this case, the total number of seeds planted).
In this scenario, we have:
- k = 4 (exactly 4 seeds don't grow),
- p = 0.35 (since the probability of a seed not growing is 1 - 0.65 = 0.35), and
- n = 11 (since 11 seeds are planted).
Now let's calculate the probability using the formula:
P(X = 4) = C(11, 4) * 0.35^4 * (1 - 0.35)^(11 - 4)
First, let's calculate the binomial coefficient:
C(11, 4) = 11! / (4! * (11 - 4)!)
= 11! / (4! * 7!)
= (11 * 10 * 9 * 8) / (4 * 3 * 2 * 1)
= 330
Now let's substitute the values into the formula:
P(X = 4) = 330 * 0.35^4 * (1 - 0.35)^(11 - 4)
Calculating the exponent terms:
0.35^4 = 0.01500625
(1 - 0.35)^7 = 0.2636625
Finally, substitute the calculated values into the formula:
P(X = 4) = 330 * 0.01500625 * 0.2636625
P(X = 4) ≈ 0.3892 (rounded to four decimal places)
Therefore, the probability that exactly 4 out of 11 seeds don't grow is approximately 0.3892.