Question

If $A$ and $B$ are two events such that $P\left( A \right) = 0.8$, $P\left( B \right) = 0.6$, and $P\left( {A \cap B} \right) = 0.5$. Then what is the value of $P\left( {A|B} \right)$?
A. $\dfrac{5}{6}$
B. $\dfrac{5}{8}$
C. $\dfrac{9}{{10}}$
D. None of these

Answer 1

Hint: Use the formula of the conditional probability of event $A$ occurring when event $B$ has occurred to reach the required answer.

Formula Used:
$P\left( {A|B} \right) = \dfrac{{P\left( {A \cap B} \right)}}{{P\left( B \right)}}$

Complete step by step solution:
Given: $A$ and $B$ are two events.
$P\left( A \right) = 0.8$, $P\left( B \right) = 0.6$, and $P\left( {A \cap B} \right) = 0.5$

Let’s calculate the probability of the event $\left( {A|B} \right)$.
$\left( {A|B} \right)$ means event $A$ is occurring given that event $B$ has already occurred.
Now apply the formula of conditional probability.
$P\left( {A|B} \right) = \dfrac{{P\left( {A \cap B} \right)}}{{P\left( B \right)}}$
Substitute the given values.
$P\left( {A|B} \right) = \dfrac{{0.5}}{{0.6}}$
To remove the decimal point, multiply the numerator and denominator by $10$.
$P\left( {A|B} \right) = \dfrac{5}{6}$

Option ‘A’ is correct

Note: Probability means how likely something is to happen. The probability of an event lies between 0 and 1.
Conditional Probability is the probability of an event $E$ occurring when the event $F$ has already occurred.
Formula: $P\left( {E|F} \right) = \dfrac{{P\left( {E \cap F} \right)}}{{P\left( F \right)}}$
The denominator is the probability of an already occurred event.