Question

A breeder of show dogs is interested in the number of female puppies in a litter. If a birth is equally likely to result in a male or a female puppy, give the probability distribution of the variable \(x=\) number of female puppies in a litter of size 5 .

Short answer

The probability distribution of \( x \) = number of female puppies in a litter of size 5 is given by: \( P(X=k) \) for \( k=0 \) to \( 5 \), calculated using the binomial distribution formula.

Step-by-step solution

Prepare your Inputs

First, identify values for n, p, and the possible values for k. Here, n = 5 (litter size), p = 0.5 (probability of getting a female), and k varies from 0 to 5 (count of females in the litter). Therefore, we need to calculate probabilities for k=0, k=1, k=2, k=3, k=4, and k=5.

Use Binomial Distribution Formula

Apply the binomial distribution formula for k=0 to 5. The formula is given by: \[ P(X=k) = C(n, k) \cdot (p^k) \cdot ((1-p)^{n-k}) \] where \( C(n, k) \) is the number of combinations of n items taken k at a time, calculated as \[ n! / (k!(n-k)!) \] Here, '!' denotes the factorial of a number.

Calculate Individual Probabilities

Calculate the individual probabilities for k=0 to 5. These will form the probability distribution of x.

Ensure Probabilities Add to 1

Finally, ensure that all of these individual probabilities add up to 1. This is a characteristic of any probability distribution.