21% of a certain breed of rabbits are born with long hair. What is the probability that in a litter of 12 rabbits, exactly 2 will have long hair? (Give your answer correct to three decimal places.)
1=.21=.79
0.79^12=0.05909
2-0.0591=1.94
Prob(long hair) = .21
prob(not long hair) = .79
prob(exactly 2 of 12 to have long hair)
= C(12,2) (.21^2)(.79^10) = appr .2756
0.05909 😀😀😀
To find the probability that exactly 2 rabbits in a litter of 12 will have long hair, we need to use the binomial probability formula. Let's break down the steps:
1. The probability of a rabbit being born with long hair is 21%, which can be expressed as 0.21. Therefore, the probability of a rabbit not having long hair is 1 - 0.21 = 0.79.
2. The probability of exactly 2 rabbits out of 12 having long hair can be calculated using the binomial probability formula:
P(X = k) = (nCk) * p^k * (1 - p)^(n - k),
where n is the total number of trials or rabbits in this case (12), k is the number of successes (2), p is the probability of success (0.21) and (nCk) denotes the binomial coefficient which calculates the number of possible combinations of selecting k items out of n.
The formula becomes:
P(X = 2) = (12C2) * (0.21^2) * (0.79^(12-2)) = 66 * 0.0441 * 0.2587 ≈ 0.7490.
Hence, the probability that exactly 2 rabbits out of 12 will have long hair is approximately 0.749, rounded to three decimal places.
To calculate the probability that exactly 2 out of 12 rabbits will have long hair, we can use the binomial probability formula.
The formula for binomial probability is:
P(X=k) = (nCk) * p^k * (1-p)^(n-k)
Where:
P(X=k) is the probability of getting exactly k successful outcomes
n is the total number of trials
k is the desired number of successful outcomes
p is the probability of success in a single trial
In this case, we want to find the probability that exactly 2 out of 12 rabbits will have long hair. Given that 21% of the breed have long hair, the probability of success (p) is 0.21.
Using the formula, we can plug in the values:
P(X=2) = (12C2) * 0.21^2 * (1-0.21)^(12-2)
To calculate (12C2), we can use the formula for combinations:
(12C2) = 12! / (2! * (12-2)!)
= 12! / (2! * 10!)
Now, let's calculate each part separately:
(12C2) = 12 * 11 / (2 * 1) = 66
0.21^2 = 0.0441
(1-0.21)^(12-2) = 0.79^10 = 0.121
Putting all the values together:
P(X=2) = 66 * 0.0441 * 0.121
= 0.05909
Therefore, the probability that exactly 2 out of 12 rabbits will have long hair is approximately 0.059, rounded to three decimal places.