Question

If A and B are events with P(A) = 0.5, P(A OR B) = 0.65, P(A AND B) = 0.15, find P(B).

Answer 1

Answer:

P(B) = 0.30

Step-by-step explanation:

This is a probability problem that can be modeled by a diagram of Venn.

We have the following probabilities:

$P(A) = P_{A} + P(A \cap B) = 0.50$

In which $P_{A}$ is the probability that only A happens.

$P(B) = P_{B} + P(A \cap B) = P_{B} + 0.15$

To find P(B), first we have to find $P_{B}$, that is the probability that only B happens.

Finding $P_{B}$:

The problem states that P(A OR B) = 0.65. This is the probability that at least one of this events happening. Mathematically, it means that:

$1) P_{A} + P(A \cap B) + P_{B} = 0.65$

The problem states that P(A) = 0.5 and $P(A \cap B) = 0.15$. So we can find $P_{A}$.

$P(A) = P_{A} + P(A \cap B)$

$0.5 = P_{A} + 0.15$

$P_{A} = 0.35$

Replacing it in equation 1)

$P_{A} + P(A \cap B) + P_{B} = 0.65$

$0.35 + 0.15 + P_{B} = 0.65$

$P_{B} = 0.65 - 0.35 - 0.15$

$P_{B} = 0.15$

Since

$P(B) = P_{B} + P(A \cap B)$

$P(B) = 0.15 + 0.15$

$P(B) = 0.30$