Question

Of all airline flight requests received by a certain ticket broker, \(70 \%\) are for domestic travel \((\mathrm{D})\) and \(30 \%\) are for international flights (I). Define \(x\) to be the number that are for domestic flights among the next three requests received. Assuming independence of successive requests, determine the probability distribution of \(x\). (Hint: One possible outcome is DID, with the probability \((0.7)(0.3)(0.7)=0.147 .)\)

Short answer

The probability distribution of x is {0:0.027, 1:0.189, 2:0.441, 3:0.343}

Step-by-step solution

Identify the Parameters for the Binomial Distribution

For the Binomial Distribution, we will need two parameters: the probability of success (p) and the number of trials (n). In this case, 'success' is defined as a request for domestic flight. According to the problem, p for domestic travel (D) is 70\%, or 0.7 and the number of trials n is the three future flights for which we're trying to predict the number of domestic requests.

Determine Probability when x=0

This is the case when no domestic flight is requested in the next three requests, i.e. all are international flights. Using the formula of Binomial distribution which is \[P(x) = C(n, x) * p^x * q^{(n - x)}\], where q is the probability of not getting a domestic flight (i.e., getting an international), C is combination, p is the probability of getting a domestic flight (0.7), and x is the number of domestic flights. So for x=0, \[P(0) = C(3, 0) * (0.7)^0 * (0.3)^3 = 1 * 1 * 0.027 = 0.027\]

Determine Probability when x=1 (one successful)

The calculation will be similar to Step 2 but x is 1. \[P(1) = C(3, 1) * (0.7)^1 * (0.3)^2 = 3 * 0.7 * 0.09 = 0.189\]

Determine Probability when x=2 (two successes)

\[P(2) = C(3, 2) * (0.7)^2 * (0.3)^1 = 3 * 0.49 * 0.3 = 0.441\]

Determine Probability when x=3 (three successes)

\[P(3) = C(3, 3) * (0.7)^3 * (0.3)^0 = 1 * 0.343 * 1 = 0.343\].

Summarize the Probabilities in the Probability Distribution Table.

The probability distribution of x (number of domestic flight requests) is as follows: P(0) = 0.027, P(1) = 0.189, P(2) = 0.441, P(3) = 0.343.