Question

$P\left(A\right)=p,P\left(B\right)=2p,P\left(\text{Exactly one out of A and B occurs}\right)=\frac{5}{9}$. What will be the maximum value of $p$?

Answer 1

Determine the maximum value of $p$

Given $P\left(A\right)=p$and $P\left(B\right)=2p$

Using the concept of probability which states the term "probability" simply refers to the likelihood of something occurring. We can talk about the probabilities of certain outcomes—how likely they are—when we're unsure about the outcome of an event. Statistics is the study of occurrences guided by probability.

We know that: $P(A\text{'}\cap B)+P(B\text{'}\cap A)=\frac{5}{9}$

Hence

$$\begin{gathered} P\left(A\right)-P(A\cap B)+P\left(B\right)-P(A\cap B)=5/9 \\ 3p-2P(A\cap B)=5/9 \\ 0\le P(A\cap B)\le p \end{gathered}$$

For maximum value $3p-2p=\frac{5}{9}$

Therefore $P=\frac{5}{9}$