Question

The probability that a certain person will buy a shirt is $ 0.2, $ the probability that he will buy a trouser is $ 0.3, $ and the probability that he will buy a shirt given that he buys a trouser is $ 0.4. $ Find the probability that he will buy both a shirt and a trouser. Find also the probability that he will buy a trouser given that he buys a shirt.

Answer 1

Hint: The conditional probability can be defined as the likelihood of an event or the outcome occurring which is based on the occurrence of the previous event or the outcomes. Here we will use the formula of the conditional probability.

Complete step-by-step answer:
Let us consider that the event of buying a shirt is represented by “S” and the event of buying the trouser be represented by “T”
Given that –
The probability of buying the shirt is $ P(S) = 0.2 $
And the probability of buying the trouser is $ P(T) = 0.3 $
Also, given that the probability that person will buy a shirt given that he buys a trouser is $ P(S/T) = 0.4 $
Now, using the conditional probability –
 $\Rightarrow P(S/T) = \dfrac{{P(S \cap T)}}{{P(T)}} $
Do-cross multiplication and make the unknown term the subject –
\[P(S/T) \times P(T) = P(S \cap T)\]
It can be re-written as –
\[\Rightarrow P(S \cap T) = P(S/T) \times P(T)\]
Place the values in the above equation –
\[P(S \cap T) = (0.4) \times (0.3)\]
Simplify the above equation –
\[P(S \cap T) = (0.12)\] ..... (I)
Now, the probability that person will buy a trouser given that he buys a shirt is
 $\Rightarrow P(T/S) = \dfrac{{P(S \cap T)}}{{P(S)}} $
Place the values in the above equation –
 $\Rightarrow P(T/S) = \dfrac{{0.12}}{{0.2}} $
Simplify the above fraction –
 $ \Rightarrow P(T/S) = 0.6 $
Hence, the probability that he will buy a trouser given that he buys a shirt is $ 0.6 $
So, the correct answer is “ $ 0.6 $ ”.

Note: Probability is the state of being probable and the extent to which something is likely to happen in the particular situations or the favourable outcomes. The probability of any event always ranges between zero and one. It can never be the negative number or the number greater than one.