Question

How do you find conditional probability \[?\]

Answer 1

Hint : The formula for finding conditional probability is derived from the probability multiplication rule, \[P(AandB)=P(A)\times P(B|A)\]. You may have also seen this \[P(A\bigcup{B})\]. This Union Symbol means “and”, as in event \[A\]happening and event \[B\] happening.

Complete step-by-step solution:
Before knowing how to find the Conditional probability, first we should take a look at what is Conditional Probability, when to apply it and from where the formula for finding conditional probability derived, so that we will be able to think more logically for such a condition by not depending on formulas only.
To understand it, we can see first the example of Unconditional Probability:
Example: Experiment is Tossing two coins,
Let say we are focusing on events:
\[A\]: Getting two heads
\[B\]: At least one head
Lets call Head as \[H\] and Tail as \[T\].
So we will have sample space of outcomes: \[\left\{ HH,TT,HT,TH \right\}\]
So by using basic formula of probability we will have:
\[P(A)=\dfrac{1}{4}\]
\[P(B)=\dfrac{3}{4}\]
Above both events were independent of each other.
But now what conditional probability says is:
\[P(A|B)\] i.e. it can be called by Probability of \[A\]stroke \[B\] or \[A\]By \[B\], which means probabilty of \[A\] where event \[B\] already occurred.
If we take above same example and find \[P(A|B)\] i.e. we will find Probability of getting two heads where event of getting at least one head already occurred.
So, logically we will have sample space left with us after event \[B\] occurred: \[\left\{ HH,HT,TH \right\}\]
\[P(Getting\_two\_heads)=\dfrac{Number\_of\_favourable\_outcomes}{Total\_outcomes}\]
                                                \[=\dfrac{1}{3}\]
So this way you can find the conditional probability that is by reducing the sample space according to the event already occurred and then find required probability.
The other very common way to find it is by the formula which is derived from the probability multiplication rule, \[P(AandB)=P(A)\times P(B|A)\]
Formula is \[P(A|B)=\dfrac{P(A\bigcap{B})}{P(B)}\] where \[P(B)\ne 0\]
If we apply this formula in above example i.e. \[P(getting\_two\_heads|at\_least\_one\_heads)=\dfrac{P(getting\_2\_H\bigcap{at\_least\_1\_H})}{P(at\_least\_1\_H)}\]
\[P(A|B)=\dfrac{\dfrac{1}{4}}{\dfrac{3}{4}}\]
\[P(A|B)=\dfrac{1}{3}\]
So, we got the same answer logically and by formula also.
Hence, this way you can find the conditional probability.

Note:In everyday situations, conditional probability is used. For example, finding a probability of a team scoring better in the next match as they have a former Olympian for a coach is conditional probability compared to the probability when a random player is hired as a coach.