Question

Use the information that, for events \(\mathrm{A}\) and \(\mathrm{B},\) we have \(P(A)=0.4, P(B)=0.3,\) and \(P(A\) and \(B)=0.1.\) Find \(P(A\) or \(B).\)

Short answer

The probability of either event A or event B happening, denoted \(P(A \cup B)\), is 0.6.

Step-by-step solution

Understand the Given Probabilities

The problem gives us the probability of Event A, \(P(A) = 0.4\), the probability of Event B, \(P(B) = 0.3\), and the probability of both Event A and Event B occurring, \(P(A \cap B) = 0.1\). These are the main pieces of information necessary to solve the problem.

Utilize the Formula for the Union of Two Events

We need to find the probability of A or B, denoted as \(P(A \cup B)\). The formula for this is given as \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). Using the given probabilities from the problem, we substitute these values into the equation.

Substituting the Given Values

Substituting the given values into the equation results in the following: \(P(A \cup B)= 0.4 + 0.3 - 0.1 = 0.6\). This indicates that the probability that either event A occurs, or event B occurs, or both occur, is 0.6.