Question

Of all airline flight requests received by a certain discount ticket broker, \(70 \%\) are for domestic travel (D) and \(30 \%\) are for international flights (I). Let \(x\) be the number of requests among the next three requests received that are for domestic flights. Assuming independence of successive requests, determine the probability distribution of x. (Hint: One possible outcome is DID, with the probability \((.7)(.3)(.7)=.147 .\) )

Short answer

The probability distribution of \(x\) (the number of Domestic Flight requests among the next three requests), using a binomial model, gives the probabilities of the four possible outcomes: \(x = 0\), \(x = 1\), \(x = 2\), \(x = 3\).

Step-by-step solution

Analyze the components for binomial distribution

In this problem, the number of trials (n) can be determined as 3. Each trial is independent, and it results in a success with probability \(p = 0.7\) (Domestic flight request) and a failure with probability \(q = 0.3\) (International flight request). The random variable \(x\) is the number of successes (number of Domestic Flight requests received), which can be 0, 1, 2, or 3

Formulate the binomial distribution

We can use the binomial distribution formula to find the probability of getting x successes in n trials. The binomial distribution formula is \(P(X = x) = {n \choose x} p^x (1-p)^{n-x}\), where \(P(X = x)\) is the probability that exactly x successes occur in n trials, \(p\) is probability of success in a single trial, \(n\) is the number of trials, and \({n \choose x}\) is the number of combinations of n items taken x at a time.

Calculate the probabilities

Here, \(x\) can take values 0, 1, 2, or 3, which means the probability distribution must include these four possible outcomes. Using the formula from step 2, calculate the probability for each outcome. For example, the probability of getting exactly 1 Domestic Flight request (\(x = 1\)) in 3 trials is \(P(X = 1) = {3 \choose 1} (0.7)^1 (1-0.7)^{3-1}\). Do the same calculation for \(x = 0\), \(x = 2\), \(x = 3\) to obtain the full probability distribution.