Problem 64 - A Complete Resource Book in Mathematics for JEE Main

Question

Consider 5 independent Bernoulli's trials each with probability of success \(p .\) If the probability of at least one failure is greater than or equal to \(31 / 32\), then \(p\) lies in the interval (A) \(\left(\frac{11}{12}, 1\right)\) (B) \(\left(\frac{1}{2}, \frac{3}{4}\right]\) (C) \(\left(\frac{3}{4}, \frac{11}{12}\right]\) (D) \(\left[0, \frac{1}{2}\right]\)

Step-by-Step Solution

Answer

The interval for \(p\) is (B) \(\left(\frac{1}{2}, \frac{3}{4}\right]\).

Step 1: Understand the Problem

We have 5 independent Bernoulli trials, each with probability of success \(p\). We need to find the interval in which \(p\) lies, given that the probability of at least one failure is greater than or equal to \(\frac{31}{32}\).

Step 2: Express the Probability of Failure

Each trial has a probability of failure equal to \(1-p\). Therefore, the probability of failure for no trials (all are successes) is \((1-p)^5\).

Step 3: Set Up the Inequality

We are given that the probability of at least one failure is greater than or equal to \(\frac{31}{32}\). The complement (all 5 trials succeed) has probability \((1-p)^5\). So, solve \(1 - (1-p)^5 \geq \frac{31}{32}\).

Step 4: Solve the Inequality

First, rearrange the inequality: \( (1-p)^5 \leq \frac{1}{32} \). Next, take the fifth root: \[1-p \leq \left(\frac{1}{32}\right)^{1/5}\]Calculate \((1/32)^{1/5} = 1/2\), so \(1-p \leq 1/2\).

Step 5: Determine the Interval for p

From the inequality \(1-p \leq 1/2\), solve for \(p\):\[p \geq 1/2\]Thus, the interval for \(p\) is \([1/2, 1]\). Since the options are \( \frac{1}{2} \leq p \leq \frac{3}{4} \) in option (B), from this interval, the solution is (B) \(\left(\frac{1}{2}, \frac{3}{4}\right]\).

Key Concepts

Probability of Success

In the context of Bernoulli trials, the probability of success is denoted by the symbol \( p \). This represents the chance that a single trial, or experiment, results in the desired outcome. For example, if you are flipping a coin, and you define getting heads as success, then \( p \) would be 0.5 for a fair coin.

When considering multiple trials, this probability remains constant across each trial. Thus, in this exercise with 5 trials, every single trial has the same probability \( p \). Recognizing this constant probability is crucial because it forms the basis for determining other probabilities, such as the probability of failure or the probability of a specific number of successes or failures across trials.

Probability of Failure

In a Bernoulli trial scenario, the probability of failure is simply the complement of the probability of success. Mathematically, it is represented as \( 1 - p \).

If each trial is independent, meaning the outcome of one does not affect another, the probability of failing across all trials needs multiplication. For instance, the probability of failing in all 5 independent trials is calculated as \((1-p)^5\). This formula is used in the exercise to find the chance of getting at least one failure.

It's important to understand that this probability is also consistent across trials, just like the probability of success. Thus, each trial individually having a failure probability of \(1-p\) will overall affect the probability of events where multiple trials are considered together.

Independent Trials

In probability, independent trials refer to experiments where the outcome of one does not influence or alter the outcomes of subsequent ones. This concept is at the heart of Bernoulli trials.

For example, if you toss a coin five times, each toss is independent. The outcome of the first toss, whether a head or a tail, does not change the probability of getting a head or a tail on the second toss. In terms of formulae, this means that the probability of a certain sequence of successes and failures is simply the product of the probabilities of each result happening individually.

In this problem, independence allows the use of powers in calculating the probability of all trials failing, as shown with \((1-p)^5\). Recognizing independence can significantly simplify complex probability calculations.

Inequality Solving

Inequality solving in probability problems often involves setting up the right inequality statement and then manipulating it to find certain unknowns, like the probability \( p \) in this exercise.

In this specific case, the inequality is based on the complement rule of probability. We know that the probability of at least one failure out of five trials equals \( 1 - (1-p)^5 \) and is given to be greater than or equal to \( \frac{31}{32} \). By solving \( (1-p)^5 \leq \frac{1}{32} \), and then applying mathematical operations like taking the fifth root, we determine the maximum allowable probability \( 1-p \), which thus reveals the lowest possible value for \( p \) through rearrangement.

This step-by-step mathematical manipulation is a common method in solving inequalities involving probabilities, enabling the deduction of certain interval values for chance events under specified conditions.