Question

Given that E and F are events such that $P\left( E \right)=0.6,P\left( F \right)=0.3,P\left( E\cap F \right)=0.2$, find $6P\left( F|E \right)$.

Answer 1

Hint: Use the formula for calculating the conditional probability of two given events which is $P\left( F|E \right)=\dfrac{P\left( E\cap F \right)}{P\left( E \right)}$ and substitute the values of given probability of events.

We have two events $E$ and $F$ such that $P\left( E \right)=0.6,P\left( F \right)=0.3,P\left( E\cap F \right)=0.2$. We have to find the value of $6P\left( F|E \right)$.
We will first evaluate the value of the conditional probability $P\left( F|E \right)$ which is the probability of occurrence of event $F$ given that the event $E$ has already occurred.
We will use the formula for conditional probability which says that $P\left( F|E \right)=\dfrac{P\left( E\cap F \right)}{P\left( E \right)}$.
Substituting the values $P\left( E \right)=0.6,P\left( E\cap F \right)=0.2$ in the above formula, we get $P\left( F|E \right)=\dfrac{P\left( E\cap F \right)}{P\left( E \right)}=\dfrac{0.2}{0.6}=\dfrac{2}{6}=\dfrac{1}{3}$.
Thus, we have $P\left( F|E \right)=\dfrac{1}{3}$.
We now have to calculate $6P\left( F|E \right)$. Thus, we have $6P\left( F|E \right)=6\left( \dfrac{1}{3} \right)=2$.
Hence, we have $6P\left( F|E \right)=2$.
Probability of any event describes how likely an event is to occur or how likely it is that a proposition is true. The value of probability of any event always lies in the range $\left[ 0,1 \right]$ where having $0$ probability indicates that the event is impossible to happen, while having probability equal to $1$ indicates that the event will surely happen. We must remember that the sum of probability of occurrence of some event and probability of non-occurrence of the same event is always $1$.

Note: Conditional probability is a measure of the probability of occurrence of an event given that another event has occurred. $P\left( A|B \right)$ measures the occurrence of event $A$ given that event $B$ has already occurred. If $A$ and $B$ are two independent events (which means that the probability of occurrence or non-occurrence of one event doesn’t affect the probability of occurring or non-occurring of the other event), then $P\left( A|B \right)$ is simply the probability of occurrence of event $A$, i.e. $P\left( A \right)$.