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Chapter 7: Problem 50

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

Expert verified
The probability distribution of the number of female puppies in a litter of size 5 is given by applying the binomial distribution formula for each possible outcome.

Step by step solution

01

Understanding the problem

The problem is about finding the probability distribution for a binomial experiment. The binomial experiment in this case is the birth of puppies with a size of 5, with a successful outcome defined as the birth of a female puppy. The probability of each outcome (male or female) is equal, therefore the probability of success is 0.5. The random variable \(x\) represents the number of successful outcomes, namely the number of female puppies in a litter of 5.
02

Apply the binomial distribution formula

To find the probability distribution of the variable, use the binomial distribution formula: \(P(X=k) = C(n,k) \cdot p^k \cdot (1-p)^{n-k}\). In this formula \(n\) represents the number of trials (5 in this case), \(p\) represents the probability of success (0.5 in this case), \(k\) represents the number of successful outcomes (number of female puppies), and \(C(n,k)\) is the binomial coefficient representing the number of combinations of \(n\) items taken \(k\) at a time. Use this formula to calculate the probability for each possible value of \(k\) (from 0 to 5).
03

Calculate the probabilities

Calculate the probability for each possible value of \(k\), from 0 to 5. This will give the probability distribution of the number of female puppies in a litter of size 5.

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