Question

A package in the mail can either be lost, delivered or damaged while being delivered. If the probability of loss is \(.2\), the probability of damage is \(.1\) and the probability of delivery is \(.7\) and 10 packages are sent to Galveston, Texas, what is the probability that 6 arrive safely 2 are lost and 2 are damaged?

Short answer

The probability that 6 packages arrive safely, 2 are lost, and 2 are damaged is approximately 6.01%. We calculate this using the multinomial probability formula, which takes into account the total number of packages, probabilities of each outcome, and the desired number of occurrences for each outcome.

Step-by-step solution

Understand the multinomial probability formula

The multinomial probability formula is an extension of the binomial probability formula and is used when we have more than two different outcomes in an experiment. In our case, we have 3 different outcomes - package lost (L), package damaged (D), and package safely delivered (S). The general formula for multinomial probability is: \(P(X_1=n_1,X_2=n_2,\dots,X_k=n_k)=\frac{n!}{n_1!n_2!\dots n_k!}p_1^{n_1}p_2^{n_2}\dots p_k^{n_k}\) where n is the total number of trials (packages), \(n_i\) is the number of occurrences of each outcome, and \(p_i\) is the probability of each outcome. In our case, we have: - n = 10 packages - \(n_1\) = 6 safely delivered packages - \(n_2\) = 2 lost packages - \(n_3\) = 2 damaged packages - \(p_1\) = 0.7 (probability of safe delivery) - \(p_2\) = 0.2 (probability of loss) - \(p_3\) = 0.1 (probability of damage)

Calculate the probabilities

Now that we know the values of each variable in the multinomial formula, we can plug in the values and calculate the probability: \(P(S=6,L=2,D=2)=\frac{10!}{6!2!2!}0.7^6 0.2^2 0.1^2\) To calculate the factorials: - \(10!=3628800\) - \(6!=720\) - \(2!=2\) Plug these values back into the formula: \(P(S=6,L=2,D=2)=\frac{3628800}{720\cdot2\cdot2}0.7^6 0.2^2 0.1^2 = 127008\) Now calculate the probability of each outcome raised to the power of its occurrences: - \(0.7^6 = 0.117649\) - \(0.2^2 = 0.04\) - \(0.1^2 = 0.01\) Plug-in these values back into the formula: \(P(S=6,L=2,D=2)=127008\cdot0.117649\cdot0.04\cdot0.01=0.060130\)

Interpret the result

Thus, the probability that 6 packages arrive safely, 2 are lost, and 2 are damaged is approximately 6.01%.