sue, a botanist, is studying a new plant seed. if it is known that these seeds have a 95% probability of germinating. if sue plants seeds, what is the probability that; a) exactly four seeds will germinated? b) at least five will germinated? c) at most one will not germinated?
We need to know how many she planted.
Given n, use binomial distribution with p = 0.95 and 1-p = 0.05
To calculate the probabilities, we can use the binomial probability formula, which is:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) represents the probability of exactly k seeds germinating.
- (n C k) is the binomial coefficient, calculated as n! / (k!(n - k)!). It represents the number of ways to choose k items out of a total of n items.
- p is the probability of germination for each seed.
- k is the number of seeds germinating.
- n is the total number of seeds.
Now, let's calculate the probabilities for each scenario:
a) Exactly four seeds will germinate:
For this scenario, n = total number of seeds = 1 (since Sue is studying a single seed), k = 4, and p = probability of germination = 0.95.
P(X = 4) = (1 C 4) * 0.95^4 * (1 - 0.95)^(1 - 4)
= 0 * 0.95^4 * (1 - 0.95)^(-3)
= 0
The probability is 0 because it's impossible to have four germinated seeds out of a single seed.
b) At least five seeds will germinate:
For this scenario, n = 1, k = 5, 6, 7, 8, ... (since we're looking for "at least" five), and p = 0.95.
P(X >= 5) = P(X = 5) + P(X = 6) + P(X = 7) + ...
We can use the complement rule to calculate P(X >= 5) as 1 - P(X <= 4).
P(X <= 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
P(X = 0) = (1 C 0) * 0.95^0 * (1 - 0.95)^(1 - 0)
= 1 * 1 * 0 = 0
P(X = 1) = (1 C 1) * 0.95^1 * (1 - 0.95)^(1 - 1)
= 1 * 0.95 * 1 = 0.95
P(X = 2) = (1 C 2) * 0.95^2 * (1 - 0.95)^(1 - 2)
= 0 * 0.95^2 * 0 = 0
P(X = 3) = (1 C 3) * 0.95^3 * (1 - 0.95)^(1 - 3)
= 0 * 0.95^3 * (-1)^2 = 0
P(X = 4) = (1 C 4) * 0.95^4 * (1 - 0.95)^(1 - 4)
= 0 * 0.95^4 * (1 - 0.95)^(-3)
= 0
P(X <= 4) = 0 + 0.95 + 0 + 0 + 0 = 0.95
P(X >= 5) = 1 - P(X <= 4)
= 1 - 0.95
= 0.05
So, the probability that at least five seeds germinate is 0.05.
c) At most one seed will not germinate:
For this scenario, n = 1, k = 0, 1 (since we're looking for "at most" one), and p = 0.95.
P(X <= 1) = P(X = 0) + P(X = 1)
P(X = 0) = (1 C 0) * 0.95^0 * (1 - 0.95)^(1 - 0)
= 1 * 1 * 0 = 0
P(X = 1) = (1 C 1) * 0.95^1 * (1 - 0.95)^(1 - 1)
= 1 * 0.95 * 1 = 0.95
P(X <= 1) = 0 + 0.95 = 0.95
So, the probability that at most one seed does not germinate is 0.95.