If a seed is planted, it has a 95% chance of growing into a healthy plant.
If 6 seeds are planted, what is the probability that exactly 4 don't grow?
To calculate the probability of exactly 4 seeds not growing, we need to use the Binomial Probability Formula. The formula is:
P(X=k) = (nCk) * (p^k) * (1-p)^(n-k)
where:
P(X=k) is the probability of getting exactly k successes (in this case, seeds not growing)
n is the number of trials (seeds planted)
k is the number of desired successes (seeds not growing)
p is the probability of success (seed not growing in this case)
Given:
n = 6 (seeds planted)
k = 4 (seeds not growing)
p = 0.05 (probability of not growing, 1 - 0.95)
Using the formula, we can substitute the values:
P(X=4) = (6C4) * (0.05^4) * (1-0.05)^(6-4)
Now, let's calculate this step-by-step:
Step 1: Calculate the binomial coefficient (6C4):
(6C4) = 6! / (4! * (6-4)!)
= 6! / (4! * 2!)
= (6 * 5 * 4!) / (4! * 2)
= 6 * 5 / 2
= 15
Step 2: Calculate the probability of getting exactly 4 seeds not growing (P(X=4)):
P(X=4) = (6C4) * (0.05^4) * (1-0.05)^(6-4)
= 15 * (0.05^4) * (0.95^2)
≈ 0.000796883
Therefore, the probability that exactly 4 seeds don't grow is approximately 0.000796883, or about 0.08%.
To find the probability that exactly 4 seeds don't grow, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes,
- n is the number of trials (in this case, the number of seeds planted),
- k is the number of successes (in this case, the number of seeds that don't grow),
- p is the probability of success in a single trial (the probability that a seed doesn't grow),
- (nCk) is the binomial coefficient (the number of ways to choose k successes from n trials).
In this case:
- n = 6 (6 seeds planted),
- k = 4 (4 seeds don't grow),
- p = 0.05 (the probability that a seed doesn't grow, which is 1 - 0.95).
Now, let's calculate the probability:
P(X = 4) = (6C4) * (0.05)^4 * (1-0.05)^(6-4)
To calculate the binomial coefficient (6C4), we use the formula:
(nCk) = n! / (k! * (n-k)!)
In this case, (6C4) = 6! / (4! * (6-4)!) = 6! / (4! * 2!) = (6 * 5) / (2 * 1) = 15.
Substituting these values into the formula, we have:
P(X = 4) = 15 * 0.05^4 * (1-0.05)^(6-4)
Calculating this value, we find:
P(X = 4) ≈ 0.002957
Therefore, the probability that exactly 4 seeds don't grow is approximately 0.002957, or 0.2957%.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
(.95^2)(.05^4) = ?