Question

a stockbroker knows from experience that the probability that a client owns stocks is 0.60. p(a)
the probability that a client owns bonds is 0.50. p(b)
the probability that the client owns bonds if he/she already owns stock is 0.55
given that the client owns bonds, what is the probability that the client owns stocks?
what is the probability of a client owning stocks or bonds?

Explanation:

Step1: Use the formula for conditional probability

The formula for conditional probability is \(P(B|A)=\frac{P(A\cap B)}{P(A)}\). We know \(P(B|A) = 0.55\), \(P(A)=0.60\). First, find \(P(A\cap B)\) by rearranging the formula: \(P(A\cap B)=P(B|A)\times P(A)\).

$$P(A\cap B)=0.55\times0.60 = 0.33$$

Step2: Use the formula for \(P(A\cup B)\)

The formula for \(P(A\cup B)\) is \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). We know \(P(A) = 0.60\), \(P(B)=0.50\), and \(P(A\cap B)=0.33\).

$$P(A\cup B)=0.60 + 0.50-0.33$$

$$P(A\cup B)=0.77$$

Answer:

\(0.77\)