Question

Suppose that 15% of a certain breed of goats are born without a tail. Find the probability that in a herd of 22 goats there are at least 6 with no tail.

Answer 1

To find the probability of at least 6 out of 22 goats being born without a tail, when 15% of them naturally lack tails, the binomial probability formula is used. One would sum the probabilities from 6 goats up to the full 22 to get the desired cumulative probability.

Step-by-step explanation:

To determine the probability that in a herd of 22 goats at least 6 are born without a tail, given that 15% of this goat breed are tailless, we can use the binomial probability formula. Since we're calculating the probability of 'at least' 6 goats, we'll have to consider all possibilities from 6 to 22.

The binomial probability formula is:

P(X = k) = (n choose k) * (p^k) * ((1-p)^(n-k))

Here, n is the total number of goats, k is the number of goats without tails that we're interested in, and p is the probability of a goat being born without a tail.

To find the total probability for at least 6 goats, we need to calculate and sum the probabilities from k=6 to k=22:

P(X ≥ 6) = P(X = 6) + P(X = 7) + ... + P(X = 22)

This is typically done using a statistical software or calculator capable of binomial distributions, as the manual calculation can be quite extensive.

Answer 2

To find the probability of having at least 6 goats without a tail in a herd of 22 goats, we can use the binomial distribution.

Step-by-step explanation:

To find the probability that in a herd of 22 goats there are at least 6 with no tail, we will use the binomial distribution.

The probability that a goat is born without a tail is 15%, so the probability that a goat has a tail is 1 - 15% = 85%.

Using the binomial probability formula, we can calculate the probability of getting exactly k goats without a tail out of n goats:

P(k successes) = C(n, k) * p^k * (1 - p)^(n-k)

We want to find the probability of having at least 6 goats without a tail. This means we need to calculate the probabilities for k = 6, 7, 8, ..., 22, and sum them up:

P(at least 6 goats without a tail) = P(6) + P(7) + P(8) + ... + P(22)