Problem 67 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A class consists of 80 students, 25 of them are girls. If 10 of the students are rich and 20 of the students are fair complexioned, then the probability of selecting a fair complexioned rich girl from the class (assuming three traits as independent) is (A) \(1 / 10\) (B) \(1 / 32\) (C) \(5 / 512\) (D) \(7 / 512\)

Step-by-Step Solution

Answer

The probability is \(\frac{5}{512}\), answer (C).

Step 1: Determine Probabilities

First, calculate the probability of each trait separately. - Probability of being a girl = \( \frac{25}{80} = \frac{5}{16} \)- Probability of being rich = \( \frac{10}{80} = \frac{1}{8} \)- Probability of being fair complexioned = \( \frac{20}{80} = \frac{1}{4} \)These probabilities are assumed to be independent.

Step 2: Calculate Combined Probability

To find the probability of a student being a fair complexioned, rich girl, multiply the individual probabilities. Thus, the probability \( P(\text{fair complexioned, rich girl}) = \frac{5}{16} \times \frac{1}{8} \times \frac{1}{4} \).

Step 3: Simplify the Probability

Simplify the resulting fraction. \[ P(\text{fair complexioned, rich girl}) = \frac{5}{16} \times \frac{1}{8} \times \frac{1}{4} = \frac{5}{512}. \]

Step 4: Determine the Correct Option

Identify which of the provided options corresponds to the calculated probability. The correct answer is (C) \(\frac{5}{512}\).

Key Concepts

Independent Events

In probability, independent events are crucial to understand because they do not affect each other's outcomes. When two events are independent, the occurrence of one event does not provide any information about the occurrence of the other.
For instance, flipping a coin and rolling a die are independent because the result of the coin flip has no bearing on the result of the die roll.

In the context of the exercise, the traits of being a girl, being rich, and being fair complexioned are considered independent. This means that knowing a student is a girl does not change the probability of her being rich or fair complexioned. In calculations involving independent events, we find the combined probability by multiplying their individual probabilities.

Multiplication Rule in Probability

The multiplication rule in probability is used to find the probability of the intersection of two or more independent events. Specifically, if events A, B, and C are independent, the probability of all three events occurring simultaneously is the product of their individual probabilities.
Formally, it's expressed as:

• \[ P(A ext{ and } B ext{ and } C) = P(A) \times P(B) \times P(C) \]

This rule was applied in the original exercise to calculate the probability of selecting a fair complexioned, rich girl.

Each of the traits - being a girl, being rich, and being fair complexioned - had their probabilities computed first. Then, the multiplication rule was brought in to determine:

• \[P( ext{fair complexioned, rich girl}) = \frac{5}{16} \times \frac{1}{8} \times \frac{1}{4} = \frac{5}{512}\]

Using this method, we computed the likelihood of all these independent events occurring together, leading us to the correct answer.

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already happened. This concept is often expressed with the formula:

• \[P(A|B) = \frac{P(A ext{ and } B)}{P(B)}\]

It is important to note that this exercise assumes the traits are independent, meaning conditional probability calculations aren't necessary here.

However, understanding conditional probability gives you a deeper insight into how probability works when events are not independent. In cases where independence cannot be assumed, the probability of one trait being present may change if you know another trait is present.
For example, knowing someone is rich might alter the probability of them also being fair complexioned, if these traits were not independent. Even though we didn't use conditional probability in solving this particular problem, it remains a valuable concept within the realm of probability.