Problem 64 - A Complete Resource Book in Mathematics for JEE Main 2017

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 is \( \left[0, \frac{1}{2}\right] \) (Option D).

Step 1: Understand the Problem

We are considering 5 independent Bernoulli trials where each trial has a probability of success denoted by \( p \). We need to find the range of \( p \) such that the probability of having at least one failure in these trials is greater than or equal to \( \frac{31}{32} \).

Step 2: Express the Complement Probability

First, let's express the probability of at least one failure in terms of the probability of all successes. The expression for the probability of all trials being successful is \( p^5 \). Therefore, the probability of at least one failure is \( 1 - p^5 \).

Step 3: Set the Inequality

Since the probability of at least one failure needs to be \( \geq \frac{31}{32} \), we set up the inequality: \[ 1 - p^5 \geq \frac{31}{32} \]

Step 4: Solve the Inequality

Rearrange the inequality to isolate \( p^5 \):\[ p^5 \leq 1 - \frac{31}{32} = \frac{1}{32} \]Now, solve for \( p \):\[ p \leq \left( \frac{1}{32} \right)^{1/5} \]

Step 5: Approximate the Fifth Root

Calculate \( \left( \frac{1}{32} \right)^{1/5} \) which gives approximately \( 0.5 \). Hence, \( p \leq 0.5 \).

Step 6: Select the Correct Interval

Review the provided options to see which interval contains values of \( p \) that are \( \leq 0.5 \). The correct interval is option \( D \): \( \left[0, \frac{1}{2}\right] \).

Key Concepts

Bernoulli Trials

In probability theory, a Bernoulli trial is a random experiment where there are only two possible outcomes: success or failure. Each trial is independent, meaning the outcome of one trial does not affect another. Consider flipping a coin, which can land as heads (success denoted by "1") or tails (failure denoted by "0"). If you perform a number of such trials, each with the probability of success as \( p \), you can analyze the likelihood of various outcomes.

• Every trial behaves according to the same probability \( p \) of success.
• Examples include coin tosses, dice rolls, and yes/no surveys.

These trials are fundamental in laying the groundwork for understanding binomial distributions and are used in estimating probabilities for more complex experiments.

Inequalities

Inequalities are mathematical expressions that show the relationship of one value being larger or smaller than another. When dealing with probabilities, inequalities help us determine the range of outcomes possible under specific conditions.
For example, if you need the probability of failing at least once in a set of Bernoulli trials to be greater than a certain value, you can set up an inequality.

• An example is: \( 1 - p^5 \geq \frac{31}{32} \), where \( p^5 \) represents all successes.
• Such inequalities help isolate and solve for the variable representing probability, \( p \).

Inequalities are crucial for deriving meaningful boundaries in probability scenarios, helping to narrow down feasible solutions that satisfy given conditions.

Probability of Failure

Probability of failure refers to the chance that an event does not succeed in an independent trial. Using Bernoulli trials, this can be framed as \( 1 - p \), where \( p \) indicates the probability of success.

• When considering multiple trials, the probability of having at least one failure can be calculated by taking 1 minus the probability of all successes (\( p^5 \) in the exercise).
• The exercise explored finding when the likelihood of at least one failure is greater than or equal to \( \frac{31}{32} \).

By setting the condition \( 1 - p^5 \geq \frac{31}{32} \), the problem ensures that the trials' potential failure meets a minimum likelihood, important in scenarios where failure might indicate a significant event, such as risk assessment or quality control.