Question

If \[4P(A) = 6P(B) = 10P(A \cap B) = 1\] , then find the value of \[P(B\left| {A)} \right.\] .
A. \[\dfrac{2}{5}\]
B. \[\dfrac{3}{5}\]
C. \[\dfrac{7}{{10}}\]
D. \[\dfrac{{19}}{{60}}\]

Answer 1

Hint: First obtain the values of $P(A),P(B),P(A \cap B)$. Then use the formula of
$P(B| {A})$ to obtain the required value.

Formula used: \[P(B\left| {A)} \right. = \dfrac{{P(A \cap B)}}{{P(A)}}\]

Complete step by step solution: The given equation is,
\[4P(A) = 6P(B) = 10P(A \cap B) = 1\]
Therefore,
\[P(A) = \dfrac{1}{4},P(B) = \dfrac{1}{6},P(A \cap B) = \dfrac{1}{{10}}\].
Now,
\[P(B\left| {A)} \right. = \dfrac{{P(A \cap B)}}{{P(A)}}\]
Substitute \[P(A) = \dfrac{1}{4}\] and \[P(A \cap B) = \dfrac{1}{{10}}\] in \[P(B\left| {A)} \right. = \dfrac{{P(A \cap B)}}{{P(A)}}\] to obtain the required value.
\[P(B\left| {A)} \right. = \dfrac{{\dfrac{1}{{10}}}}{{\dfrac{1}{4}}}\]
                 \[ = \dfrac{4}{{10}}\]
                 \[ = \dfrac{2}{5}\]

So, Option ‘A’ is correct

Additional information: Bayes' theorem is a mathematical formula used in calculating conditional probability.
The probability can be either conditional or marginal or joint.
Conditional probability: In conditional probability, the probability of an event is dependent on the probability of another event.
At least two dependent events need to apply conditional probability.
Marginal probability: The probability occurring of an event is known as the marginal probability.
Joint probability: Joint is the probability such that two events occur together at the same time.

Note: Students are used to with the formula \[P(A\left| {B)} \right. = \dfrac{{P(A \cap B)}}{{P(B)}}\], so sometimes they divide \[P(A \cap B)\] by \[P(B)\] to obtain the answer but here the question is to find \[P(B\left| {A)} \right.\]. Hence we have to divide \[P(A \cap B)\] by \[P(A)\] to get the correct answer.