the probability of a seed germinates is 1/3. If three of the seed are planted, what is the probability that only one will germinate
To find the probability that only one seed will germinate out of the three planted, we need to consider the possible outcomes.
There are three possible outcomes for each seed: germinate (G) with a probability of 1/3, or not germinate (NG) with a probability of 2/3.
To calculate the probability that only one seed will germinate, we need to calculate the probability of the desired outcome, which is one germinating seed, and multiply by the probability of the other two seeds not germinating.
The probability that one seed germinates and the other two do not is calculated as follows:
Probability = (1/3) * (2/3) * (2/3)
= 2/27
Therefore, the probability that only one seed will germinate out of the three planted is 2/27.
To find the probability that only one seed will germinate out of the three planted seeds, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of getting exactly k successes,
n is the total number of trials,
k is the number of successes,
p is the probability of success in a single trial, and
(1 - p) is the probability of failure in a single trial.
In this case, the number of trials (n) is 3, the number of successes (k) is 1, and the probability of success in a single trial (p) is 1/3. So, we can substitute these values into the formula:
P(X = 1) = (3 C 1) * (1/3)^1 * (2/3)^(3 - 1)
Now, let's calculate the probability:
P(X = 1) = (3! / (1! * (3-1)!)) * (1/3)^1 * (2/3)^(3 - 1)
= (3 / (1 * 2)) * (1/3) * (4/9)
= (3/2) * (1/3) * (4/9)
= 12/54
= 2/9
Therefore, the probability that only one seed will germinate out of the three planted seeds is 2/9.
To find the probability that only one seed will germinate out of the three planted, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of having exactly k successes
- n is the number of trials or attempts
- k is the number of successful outcomes we are interested in
- p is the probability of success on a single trial
- C(n, k) is the binomial coefficient, which represents the number of ways to choose k objects from a set of n objects.
In this case, n (the number of trials) is 3, k (the number of successful outcomes) is 1, and p (the probability of success) is 1/3.
Using the formula, we can calculate the probability as follows:
P(X = 1) = C(3, 1) * (1/3)^1 * (2/3)^(3-1)
C(3, 1) can be calculated as 3! / ((3-1)! * 1!) = 3, since there are 3 ways to choose 1 success out of 3 trials.
Substituting the values, we have:
P(X = 1) = 3 * (1/3)^1 * (2/3)^(3-1)
P(X = 1) = 3 * (1/3) * (2/3)^2
P(X = 1) = 3 * (1/3) * (4/9)
P(X = 1) = 12/27
P(X = 1) = 4/9
Therefore, the probability that only one seed will germinate out of the three planted is 4/9.