If a seed is planted, it has an 85% chance of growing into a healthy plant.if 8 seeds are planted what is the probability that exactly 3 don't grow ?
To find the probability that exactly 3 seeds don't grow out of 8, we need to use the binomial probability formula.
The binomial probability formula is given by:
P(X=k) = (nCk) * (p^k) * (q^(n-k))
Where:
P(X=k) represents the probability of getting exactly k successes,
n is the total number of trials or seeds planted,
k is the number of successes or seeds that don't grow,
p is the probability of success (plant growing),
q is the probability of failure (1 - p),
nCk is the binomial coefficient or the number of ways to choose k items from a set of n items.
Using the given information:
n = 8 (total number of seeds planted)
k = 3 (number of seeds that don't grow)
p = 0.85 (probability of a seed growing)
q = 1 - p = 1 - 0.85 = 0.15 (probability of a seed not growing)
Now we can substitute these values into the binomial probability formula:
P(X=3) = (8C3) * (0.85^3) * (0.15^(8-3))
To calculate (8C3), we can use the combination formula:
(8C3) = 8! / (3! * (8-3)!)
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
3! = 3 * 2 * 1
(8-3)! = 5!
After evaluating the combination, we can substitute the values into the formula:
P(X=3) = (8C3) * (0.85^3) * (0.15^(8-3))
= (8! / (3! * (8-3)!) ) * (0.85^3) * (0.15^5)
Calculating each part separately:
8C3 = 8! / (3! * (8-3)!)
= (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((3 * 2 * 1) * (5 * 4 * 3 * 2 * 1))
= 56
Now we can substitute this value into the formula:
P(X=3) = 56 * (0.85^3) * (0.15^5)
≈ 0.0745
Therefore, the probability that exactly 3 seeds don't grow is approximately 0.0745 or 7.45%.
To find the probability that exactly 3 seeds don't grow out of 8 planted seeds, you can use the binomial probability formula.
The binomial probability formula is given by:
P(x) = (nCx) * (p^x) * (q^(n-x))
Where:
P(x) denotes the probability of x successes,
nCx represents the number of ways to choose x items out of n,
p is the probability of success, and
q is the probability of failure, which is calculated as 1 - p.
In this case, n = 8 (total number of seeds planted), p = 0.85 (probability of a seed growing), and q = 1 - p = 1 - 0.85 = 0.15 (probability of a seed not growing).
Now substitute the values into the formula:
P(3) = (8C3) * (0.85^3) * (0.15^(8-3))
Let's calculate the individual components:
(8C3) = 8! / (3! * (8-3)!) [This represents the number of ways to choose 3 seeds out of 8.)
= (8 * 7 * 6) / (3 * 2 * 1)
= 56
(0.85^3) = 0.85 * 0.85 * 0.85
= 0.614125
(0.15^(8-3)) = 0.15^5
= 0.0000759375
Now, substitute these values back into the formula:
P(3) = (56) * (0.614125) * (0.0000759375)
≈ 0.002275
Therefore, the probability that exactly 3 seeds don't grow out of 8 planted seeds is approximately 0.002275, or 0.2275%.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
.85^5 * .15^3 = ?