31% of a certain breed of rabbits are born with long hair. What is the probability that in a litter of 8 rabbits, exactly 5 will have long hair? (Give your answer correct to three decimal places.)
To find the probability of exactly 5 out of 8 rabbits having long hair, we can 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) is the probability of getting exactly k successes (in this case, exactly 5 rabbits with long hair)
n is the number of trials or total number of rabbits in the litter (in this case, n = 8)
k is the number of successful or desired outcomes (in this case, k = 5)
p is the probability of success (in this case, p = 0.31, as 31% of the breed has long hair)
q is the probability of failure (q = 1 - p = 1 - 0.31 = 0.69)
Now, let's calculate the probability:
P(X = 5) = 8C5 * 0.31^5 * 0.69^(8-5)
To calculate 8C5, which represents the number of ways to choose 5 out of 8, we use the binomial coefficient formula:
nCk = n! / (k!(n-k)!)
So, 8C5 = 8! / (5!(8-5)!) = 56
Let's substitute these values into the formula:
P(X = 5) = 56 * 0.31^5 * 0.69^3
Calculating this expression gives:
P(X = 5) = 0.233
Therefore, the probability that exactly 5 out of 8 rabbits will have long hair is 0.233, correct to three decimal places.