if a seed is planted, it has a 70% chance of growing into a healthy plant.if 6 seeds are planted, what is the probability that exactly 1 doesn't grow?
Let event
G=seed grows into a healthy plant (success)
~G=seed does not grow (failure)
We use the binomial distribution where
N=6,
p=0.70 (probability of success)
n=5 (5 successes)
C(N,n)=N!/((N-n)!n!)
then by binomial theorem
P(n=5)=C(6,5)p^5(1-p)^1
=6!/(1!5!)0.7^5(0.3^1)
=.3025
binomial distribution
we want chance of exactly 5 of 6 making it
p(grow) = .7
so p(not) = .3
P(6,5) =C(6,5) .7^6 * .3^1
= 6 (.7)^6 * .3
= .212
To find the probability that exactly 1 seed doesn't grow out of 6, we can use the binomial probability formula:
P(x) = (nCx) * (p^x) * (q^(n-x))
Where:
P(x) is the probability of getting exactly x successes
n is the total number of trials (in this case, the number of seeds planted)
x is the number of desired successes (in this case, exactly 1 seed not growing)
p is the probability of success on a single trial (in this case, 0.7)
q is the probability of failure on a single trial (in this case, 0.3)
nCx represents the number of combinations of n items taken x at a time
Plugging in the values:
P(1) = (6C1) * (0.7^1) * (0.3^(6-1))
Calculating the combinations:
6C1 = 6 (as there are 6 ways to choose 1 seed from 6)
Simplifying the expression:
P(1) = 6 * 0.7 * (0.3^5)
Calculating the result:
P(1) = 0.30252
Therefore, the probability that exactly 1 seed doesn't grow out of 6 is approximately 0.30252 or 30.25%.
To calculate the probability that exactly 1 seed doesn't grow, we can use the binomial probability formula. The binomial probability formula is:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of exactly k successes (in this case, the seed not growing)
- n is the total number of trials (in this case, the number of seeds planted)
- k is the number of successful trials (in this case, 1 seed not growing)
- p is the probability of success for each trial (in this case, 0.30 since the seed has a 70% chance of growing)
The (n C k) term represents the number of combinations, which can be calculated as:
(n C k) = n! / (k! * (n - k)!)
Let's plug in the values and calculate the probability:
n = 6 (number of seeds planted)
k = 1 (number of seeds not growing)
p = 0.30 (probability of a seed not growing)
(n C k) = 6! / (1! * (6 - 1)!) = 6! / (1! * 5!) = 6
P(X = 1) = (6 C 1) * 0.3^1 * (1 - 0.3)^(6 - 1)
= 6 * 0.3^1 * 0.7^5
= 6 * 0.3 * 0.16807
= 0.30204
Therefore, the probability that exactly 1 seed doesn't grow is approximately 0.302 or 30.2%.