Question

# A class consists of $100$ students, $25$ of them are girls and $75$ are boys, $20$ of them are rich and remaining poor, $40$ of them are fair complexioned. The probability of selecting a fair complexioned rich girl is.

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Hint: Calculate the probability of each of the given events and use the formula for calculating the probability of intersection of independent events, which states that the probability of intersection of independent events is the product of probability of each of the individual events.

We have the data regarding a class of $100$ students. We have to find the probability of selecting a fair complexioned rich girl.
Let’s denote the event of getting a girl by $A$ and thus, the event of getting a boy is denoted by ${{A}^{c}}$.
Similarly, denote the event of getting a rich person by $B$ and thus, the event of getting a poor person is denoted by ${{B}^{c}}$.
Similarly, the event of getting a fair complexioned person is denoted by $C$ and thus, the event of getting a dark complexioned person is denoted by ${{C}^{c}}$.
We have to calculate the probability of each of these events. We know that the probability of getting any event is the ratio of the number of favourable outcomes to the total number of possible outcomes.
As there are $25$ girls in a class, we have $P\left( A \right)=\dfrac{25}{100}=\dfrac{1}{4}$ and $P\left( {{A}^{c}} \right)=\dfrac{75}{100}=\dfrac{3}{4}$.
Similarly, as we have $20$ rich students in a class, we have $P\left( B \right)=\dfrac{20}{100}=\dfrac{1}{5}$ and $P\left( {{B}^{c}} \right)=\dfrac{80}{100}=\dfrac{4}{5}$.
As there are $40$ fair complexioned students in a class, we have $P\left( C \right)=\dfrac{40}{100}=\dfrac{2}{5}$ and $P\left( {{C}^{c}} \right)=\dfrac{60}{100}=\dfrac{3}{5}$.
We now have to evaluate the probability of getting a fair complexioned rich girl, which is $P\left( A\cap B\cap C \right)$. We observe that the events $A,B,C$ are independent, which means that the occurrence or non-occurrence of one event doesn’t affect the occurrence or non-occurrence of another event.
We know that if events $A,B,C$ are independent, we have $P\left( A\cap B\cap C \right)=P\left( A \right)P\left( B \right)P\left( C \right)$.
Substituting the values from above equations, we have $P\left( A\cap B\cap C \right)=P\left( A \right)P\left( B \right)P\left( C \right)=\dfrac{1}{4}\times \dfrac{1}{5}\times \dfrac{2}{5}=\dfrac{1}{50}$.
Hence, the probability of getting a fair complexioned rich girl, denoted by $P\left( A\cap B\cap C \right)$, is equal to $\dfrac{1}{50}$.

Note: 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 $1$.