Question

A certain company sends \(40 \%\) of its overnight mail parcels by means of express mail service \(A_{1}\). Of these parcels, \(2 \%\) arrive after the guaranteed delivery time (use \(L\) to denote the event late delivery). If a record of an overnight mailing is randomly selected from the company's files, what is the probability that the parcel went by means of \(A_{1}\) and was late?

Short answer

The probability that the parcel went by means of \(A_{1}\) and was late is \(0.008\) or \(0.8\%\).

Step-by-step solution

Identify the Probabilities of Individual Events

From the problem, it's known that the probability of a parcel being sent by means of service A1, denoted as \(P(A1)\), is \(40\%\), or \(0.4\) in decimal terms. The probability of a parcel arriving late given that it was sent by A1, denoted as \(P(L | A1)\), is \(2\%\), or \(0.02\) in decimal terms.

Compute the Intersection of the Two Events

The probability of intersection of two events (i.e., the parcel being sent by A1 and arriving late) can be calculated as: \(P(A1 \cap L) = P(A1) \times P(L | A1)\).

Calculate the Result

Substituting the values into the formula, we get: \(P(A1 \cap L) = 0.4 \times 0.02 = 0.008\).