the probability of a seed germinating is 1/3.If three of the seeds are planted,what is the probability that:A.none germinates B.at least one will germinate C.only one will germinate.
A. (1 - 1/3)^3
B. "at least one" means not none ... 1 - (A. result)
C. 3C1 * 1/3 * (2/3)^2
binomial probability
A=p(germinate)=1/3
B=or(not germinate)=3/3-1/3=2/3
We have
1 3 3 1
(A+B)³=A³+3A²B+3AB²+B³
(A)=(2/3)³=8/27
B=[1-A]=[27-8]/27=19/27
C=3AB²=3(1/3)(4/9)=4/9
To calculate the probabilities, we'll use the following:
Probability of germination (p) = 1/3
Probability of no germination (q) = 1 - p = 1 - 1/3 = 2/3
A. The probability that none of the seeds germinate:
Since the probability of each seed germinating is independent, we can multiply the probabilities:
P(none germinate) = P(no germination) * P(no germination) * P(no germination)
= (2/3) * (2/3) * (2/3)
= 8/27
Therefore, the probability that none of the seeds will germinate is 8/27.
B. The probability that at least one seed will germinate:
The probability of at least one seed germinating is the complement of none germinating:
P(at least one germinates) = 1 - P(none germinate)
= 1 - (8/27)
= 19/27
Therefore, the probability that at least one seed will germinate is 19/27.
C. The probability that only one seed will germinate:
There are three possible cases where only one seed germinates. Let's calculate the probabilities for each:
P(only 1 germinates) = P(germinates) * P(no germination) * P(no germination)
+ P(no germination) * P(germinates) * P(no germination)
+ P(no germination) * P(no germination) * P(germinates)
= (1/3) * (2/3) * (2/3)
+ (2/3) * (1/3) * (2/3)
+ (2/3) * (2/3) * (1/3)
= 12/81 + 12/81 + 12/81
= 36/81
Therefore, the probability that only one seed will germinate is 36/81.
Note: The probabilities have been simplified, but you can also reduce them further if needed.
To find the probability of different outcomes related to seed germination, we can use the concept of probability. Probability is defined as the number of favorable outcomes divided by the total number of possible outcomes.
Given:
- The probability of a seed germinating is 1/3.
- Three seeds are planted.
Now, let's analyze each scenario:
A. None of the seeds germinate:
To find the probability that none of the seeds germinate, we need to calculate the probability that each seed fails to germinate and multiply them together.
Since the probability of a seed germinating is 1/3, the probability that a seed does not germinate is 1 - 1/3 = 2/3.
Since there are three seeds, the probability that none of them germinate is (2/3) * (2/3) * (2/3) = 8/27.
So, the probability that none of the seeds germinate is 8/27.
B. At least one seed germinates:
To find the probability that at least one seed germinates, we need to find the complement of the probability that all seeds fail to germinate.
The complement probability is 1 - the probability that none of the seeds germinate, which we already calculated to be 8/27.
So, the probability that at least one seed germinates is 1 - 8/27 = 19/27.
C. Only one seed germinates:
To find the probability that only one seed germinates, we need to consider three different cases: the first seed germinates while the other two don't, the second seed germinates while the other two don't, and the third seed germinates while the other two don't.
For each case, the probability is (1/3) * (2/3) * (2/3) = 4/27.
Since there are three cases, the total probability that only one seed germinates is 3 * (4/27) = 12/27.
So, the probability that only one seed germinates is 12/27.