Problem 61 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(A_{1}, A_{2}, \ldots, A_{n}\) are \(n\) independent events such that \(P(A)\) \(=\frac{1}{i+1}, i=1,2, \ldots, n\). The probability that none of the \(n\) events occurs is (A) \(\frac{n}{n+1}\) (B) \(\frac{1}{n+1}\) (C) \(\frac{n}{(n+1)(n+2)}\) (D) none of these

Step-by-Step Solution

Answer

The probability that none of the events occurs is \(\frac{1}{n+1}\) (Option B).

Step 1: Understand the Problem

We are given \(n\) independent events, \(A_1, A_2, \ldots, A_n\), where each event \(A_i\) has a probability of occurrence given by \(P(A_i) = \frac{1}{i+1}\). We need to find the probability that none of these \(n\) events occurs.

Step 2: Determine Probability of Non-Occurrence for a Single Event

For an individual event \(A_i\), the probability that it does not occur is given by \(1 - P(A_i)\). So, \(P(\text{not } A_i) = 1 - \frac{1}{i+1} = \frac{i}{i+1}\).

Step 3: Calculate Product for Independent Events

Since all events are independent, the probability that none of them occurs is the product of their individual probabilities of non-occurrence. Thus, we calculate \(P(\text{none occur}) = \prod_{i=1}^{n} \frac{i}{i+1}\).

Step 4: Simplify the Product

The product \(\prod_{i=1}^{n} \frac{i}{i+1}\) simplifies by canceling terms due to the telescopic nature:\[ \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \ldots \cdot \frac{n}{n+1} = \frac{1 \cdot 2 \cdot 3 \cdot \ldots \cdot n}{2 \cdot 3 \cdot 4 \cdot \ldots \cdot (n+1)} = \frac{1}{n+1} \].

Step 5: Choose the Correct Option

The probability that none of the events occurs is \(\frac{1}{n+1}\), which corresponds to option (B).

Key Concepts

Independent Events

In probability theory, independent events are those whose occurrence or non-occurrence does not affect each other. This means that the outcome of one event provides no information about the outcome of another event. When events are independent, the combined probability of all events happening is simply the product of their individual probabilities.

For instance, if we have two independent events \(E\) and \(F\), the joint probability that both will occur is calculated as \(P(E \text{ and } F) = P(E) \times P(F)\). This extends to any number of events, like in the original exercise where \(A_1, A_2, \ldots, A_n\) are all independent.

Understanding the concept of independent events is crucial, as it simplifies calculations. If any two events were dependent, knowing the outcome of one would alter the probability of the other, making the problem more complex.

Telescopic Product

A telescopic product is a multiplication of fractions or values where many intermediate terms cancel each other out neatly. It is very much like a telescopic series in summation, where terms far from both ends cancel out, leaving a simple result.

In the problem exercise, the probability of non-occurrence is expressed as a product of several terms: \(\prod_{i=1}^{n} \frac{i}{i+1}\). When you multiply these fractions, you'll notice that each denominator cancels out with the next numerator:

• \(\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \ldots \times \frac{n}{n+1}\)

Thus, everything cancels except for the very first numerator \(1\) and the very last denominator \(n+1\). This simplifies the entire product down to \(\frac{1}{n+1}\). This telescopic nature makes complex calculations more manageable and quick.

Non-Occurrence Probability

The non-occurrence probability is essentially about determining the chance that a specific event does not happen. For an event \(A\), if its probability of happening is \(P(A)\), then the probability of it not occurring is \(1 - P(A)\).

In the given problem, each event \(A_i\) has a chance of happening defined as \(P(A_i) = \frac{1}{i+1}\). Thus, the non-occurrence probability becomes \(1 - \frac{1}{i+1} = \frac{i}{i+1}\).

• This fraction signifies the likelihood of not seeing this event happen.
• Calculating these probabilities for each of the independent events allows the determination of the combined non-occurrence probability.

When scaled up to \(n\) independent events, the probability none of them occurs is the product of each individual non-occurrence probability, taking advantage of their independence as outlined in the telescopic product.