Problem 60 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

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 \(\frac{31}{32}\), 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

Option (b): \([0, \frac{1}{2}]\).

Step 1: Understanding the Problem

We are given 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 \( \geq \frac{31}{32} \).

Step 2: Express Probability of Failure

The probability of failure in a single Bernoulli trial is \( 1-p \). So, for 5 independent trials, the probability of all being successful is \( p^5 \).

Step 3: Calculate Probability of At Least One Failure

The probability of at least one failure in 5 trials is the complement of having all trials successful. Therefore, \( 1 - p^5 \geq \frac{31}{32} \).

Step 4: Solve the Inequality

Rearrange the inequality: \[ 1 - p^5 \geq \frac{31}{32} \]Subtract 1 from both sides:\[ -p^5 \geq \frac{31}{32} - 1 \]\[ -p^5 \geq -\frac{1}{32} \]Multiply by \(-1\) to get:\[ p^5 \leq \frac{1}{32} \].

Step 5: Find the Interval for p

Take the fifth root of both sides to solve for \( p \): \[ p \leq \left(\frac{1}{32}\right)^{1/5} \]Calculate: \[ p \leq \frac{1}{2} \]. This gives us the interval \([0, \frac{1}{2}]\).

Step 6: Select the Correct Option

The calculated interval \( [0, \frac{1}{2}] \) corresponds to option (b).

Key Concepts

Probability Theory

Probability Theory is a branch of mathematics that deals with the likelihood of events occurring. It quantifies uncertainty using numbers between 0 and 1.
For example, a probability of 0 means an event will not occur, and a probability of 1 means it will definitely occur.
Probabilities can also be expressed as fractions or percentages, making them easy to work with.In the context of Bernoulli trials, probability theory helps us understand the chances of successes or failures over a series of trials. In these trials, each event is independent, meaning the outcome of one does not affect the others.
The probability of success is denoted by \( p \), and for failure by \( 1-p \). This is particularly useful when calculating the likelihood of various outcomes. For example, if each trial has a success probability of \( p \), then the probability that all trials are successful is \( p^5 \) for five trials.

Inequalities

Inequalities are mathematical expressions used to show the relationship between two values where they are not equal. They are expressed using symbols such as \( \leq \), \( \geq \), \( < \), and \( > \).
They are essential in solving problems where the exact values are not known but can be estimated within a range.We use inequalities in probability problems to evaluate ranges of potential outcomes. For instance, in the exercise provided, the inequality \( 1 - p^5 \geq \frac{31}{32} \) represents the condition for having at least one failure in five trials. By solving inequalities, we can determine intervals where a variable, such as probability \( p \), can live. This tells us about possible and impossible scenarios for the trials. With inequalities, the result could describe a continuous range rather than a specific value.

Complementary Events

Complementary events are pairs of outcomes where the occurrence of one event means the non-occurrence of another. In probability theory, if one event happens, the complement does not, and vice-versa.
The sum of probabilities of an event and its complement is always equal to 1.In Bernoulli trials, complementary events are crucial for calculating the probability of at least one failure. Imagine you're tossing a coin several times, where getting a head is a success. The complementary event is getting tails, signifying a failure.
When expressed in formulas, if the event is success in all five trials (\( p^5 \)), then failing at least once is its complement, calculated as \( 1 - p^5 \). This is instrumental in deriving probabilities for compound trial outcomes, like calculating the probability of failure in the given problem.

Fifth Root

The fifth root of a number is the value that, when multiplied by itself five times, gives the original number. It’s represented mathematically as \( \sqrt[5]{x} \) or \( x^{1/5} \).
Understanding roots is vital for solving inequality problems that involve powers, especially like in this exercise.Taking the fifth root helps ascertain boundaries for values tightly connected with powers. For example, in the step-by-step solution, we derive the inequality \( p^5 \leq \frac{1}{32} \).
To find potential values for \( p \), we compute the fifth root: \( p \leq \left(\frac{1}{32}\right)^{1/5} \). This computation lets us express and interpret solutions related to different root levels, deriving intervals for probabilities and other variables in similar situations.