Problem 166 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

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

Step-by-Step Solution

Answer

(B) \([0, \frac{1}{2}]\)

Step 1: Identify Probability of Failure

Each Bernoulli trial has a probability of failure of \[ 1 - p. \]

Step 2: Calculate Probability of At Least One Failure

The probability of having at least one failure in 5 trials is computed using the complement rule.\[ P(\text{at least one failure}) = 1 - P(\text{zero failures}). \]

Step 3: Calculate Probability of Zero Failures

The probability of zero failures (all trials are successes) is\[ P(\text{zero failures}) = p^5. \]

Step 4: Set Up Inequality for At Least One Failure

Given the probability of at least one failure is greater than or equal to \( \frac{31}{32} \), we write:\[ 1 - p^5 \geq \frac{31}{32}. \]

Step 5: Solve the Inequality

Rearrange the inequality to solve for \( p^5 \):\[ p^5 \leq 1 - \frac{31}{32} = \frac{1}{32}. \]

Step 6: Find the Range for p

To get the range for \( p \), we solve:\[ p \leq \left(\frac{1}{32}\right)^{1/5}. \]Calculating, we find: \[ \left(\frac{1}{32}\right)^{1/5} \approx 0.5. \] Thus, \( p \leq 0.5 \).

Step 7: Determine the Correct Interval

The correct interval based on our answer that \( p \leq 0.5 \) is\[ \left[0, \frac{1}{2}\right]. \]Therefore, the correct option is (B).

Key Concepts

Bernoulli Trials

In probability theory, a Bernoulli trial is a random experiment that has exactly two possible outcomes: "success" and "failure." Each outcome is given a constant probability, which does not change between trials. Bernoulli trials are named after the Swiss mathematician Jacob Bernoulli, who studied permutations and combinations.

A Bernoulli trial can be imagined as flipping a coin where each flip (trial) results in either heads (success) or tails (failure). When dealing with a series of Bernoulli trials, each trial is independent of one another. This means the outcome of one trial does not influence the outcome of another.

• In the context of our exercise, if each trial has a probability of failure of \( 1 - p \), then the probability of success in a single trial is \( p \).
• If we consider 5 such trials, we look at the total probability cloud of these trials collectively to find the solution.

Inequality Solving

In our exercise, inequality solving involves manipulating equations to find the range of possible values for a variable that satisfies a given condition. Here, we are trying to find the range of \( p \) using inequality solving techniques.

The problem asks for the probability that at least one failure occurs out of 5 independent Bernoulli trials to be greater than or equal to \( \frac{31}{32} \). The inequality is derived from the complement rule:

• First, find the probability of zero failures, which is \( p^5 \). This represents the situation where all trials are successful.
• Then, express the inequality for at least one failure: \( 1 - p^5 \geq \frac{31}{32} \). This relation indicates that the probability of obtaining at least one failed trial must be at least \( \frac{31}{32} \).

Solving the inequality gives \( p^5 \leq \frac{1}{32} \), meaning \( p \) must be at or below the fifth root of \( \frac{1}{32} \). Calculating gives \( \left(\frac{1}{32}\right)^{1/5} \approx 0.5 \). So \( p \) should not exceed 0.5.

Complement Rule

The complement rule is a fundamental concept in probability theory. It helps in finding the probability of at least one event happening by calculating the probability of the complementary event not happening and subtracting it from 1.

In this exercise, the event of interest is "at least one failure." Its complement is "zero failures," meaning all trials are successful. This is represented as \( p^5 \) in mathematical terms.

The complement rule is then applied:

• The probability of zero failures is calculated first: \( p^5 \).
• Then, the probability of at least one failure is \( 1 - p^5 \).

This approach simplifies the problem by converting it into an equation with known components we can solve, as seen in the steps to find that \( p \) falls within a specified interval.