Question

question 11
a union executive conducted a survey of its members to determine what the membership
felt were the important issues to be resolved during upcoming negotiations with
management. the results indicate that 74% of members felt that job security was an
important issue. 45% indicated pension benefits as an important issue. 80% also felt that
considering only those who felt that pension benefits were important, 80% also felt that
job security was an important issue. what is the probability that at least one of these two
issues is important?
show all work.
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12pt paragraph b i u a - a - 2 - t -

Explanation:

Step1: Recall the formula for \(P(A\cup B)\)

The formula for the probability of the union of two events \(A\) and \(B\) is \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Let \(A\) be the event that a member feels job - security is important and \(B\) be the event that a member feels pension benefits is important. We know \(P(A) = 0.74\), \(P(B)=0.45\), and \(P(A\cap B)\) is not given directly. But we want to find \(P(A\cup B)\) (the probability that at least one of the two issues is important).

Step2: Calculate \(P(A\cup B)\)

Substitute the values into the formula: \(P(A\cup B)=0.74 + 0.45-0\) (since we are considering only those who felt that job - security or pension benefits were important. In the context of calculating the probability that at least one of the two issues is important, if we assume no double - counting adjustment is needed other than the basic formula and no information about non - overlapping in a wrong sense. Mathematically, using the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\), and if we assume that the events are non - mutually exclusive in the general probability sense. But if we consider the problem as a simple application of the formula where we are just given two probabilities of events and asked for the probability of at least one (the union). The formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\), and if we assume that we are to use the given values directly as \(P(A) = 0.74\), \(P(B)=0.45\) and since we want the probability that at least one (the union) and no other constraints on the intersection other than the formula. So \(P(A\cup B)=0.74+0.45 = 1.19\) is wrong. Wait, no, actually, we made a mistake. Let's re - do.

Let \(A\) be job - security (\(P(A)=0.74\)), \(B\) be pension - benefits (\(P(B) = 0.45\)). The probability that at least one of the two issues is important is \(P(A\cup B)\). Using the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). But if we assume that the problem is using the inclusion - exclusion principle correctly and we are given that we want \(P(A\cup B)\) (at least one). If we assume that the data is such that we can directly apply the formula. Wait, no, actually, the problem says "considering only those who felt that pension benefits were important, 80% also felt that job - security was an important issue". So \(P(A|B)=\frac{P(A\cap B)}{P(B)} = 0.8\). Then \(P(A\cap B)=P(A|B)\times P(B)=0.8\times0.45 = 0.36\)

Now, using the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)

Substitute \(P(A) = 0.74\), \(P(B)=0.45\), \(P(A\cap B)=0.36\)

\(P(A\cup B)=0.74 + 0.45-0.36\)

\(P(A\cup B)=0.83\)

Answer:

\(0.83\)