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

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

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

Step 1: Understand the Problem

We are dealing with 5 independent Bernoulli trials each with a probability of failure, given as \(1-p\). We need to find when the probability of having at least one failure is greater than or equal to \(\frac{31}{32}\).

Step 2: Express the Probability

The probability of at least one failure in 5 trials is the complement of the probability of all successes, which is \(1 - p^5\). Thus, we have the equation \(1 - p^5 \geq \frac{31}{32}\).

Step 3: Solve Inequality

To solve \(1 - p^5 \geq \frac{31}{32}\), we rewrite it as \(p^5 \leq \frac{1}{32}\). Taking the fifth root of both sides, we obtain \(p \leq \left(\frac{1}{32}\right)^{1/5}\).

Step 4: Calculate Fifth Root

Calculate \(\left(\frac{1}{32}\right)^{1/5}\). Since \(32 = 2^5\), we know \(\left(\frac{1}{32}\right)^{1/5} = \frac{1}{2}\). Therefore, \(p \leq \frac{1}{2}\).

Step 5: Determine the Interval

Given \(p \leq \frac{1}{2}\), the interval that satisfies this condition is \([0, \frac{1}{2}]\), which corresponds to option (B).

Key Concepts

Probability of Failure

In the context of Bernoulli's trials, the probability of failure is a crucial concept. A Bernoulli trial is a random experiment where there are only two possible outcomes: success or failure. For example, flipping a coin could be viewed as a Bernoulli trial, where heads might represent success and tails failure.

The exercise involves calculating the probability of failure for a series of independent trials, each with a failure probability expressed as \(1-p\). This means if something does not happen with probability \(p\), it fails with probability \(1-p\).

Understanding this concept helps in assessing the likelihood of different scenarios, especially when repeated over multiple trials. It's this likelihood or probability that guides decision-making in uncertain conditions.

Independent Trials

The concept of independent trials is fundamental in probability theory. Each Bernoulli trial's outcome is not influenced by other trials.

This independence means that knowing the result of one trial gives no information about the outcome of another. In our problem, since the trials are independent, the combined probability of successes or failures can be found by simply multiplying their respective probabilities.

In practice, if the probability of completing a task successfully is the same each time and remains constant, we are often dealing with independent trials, facilitating calculations across multiple iterations effectively.

Inequality Solving

Solving inequalities is a mathematical technique used to find a range of values that satisfy a certain condition. In the given problem, we need to solve the inequality \(1 - p^5 \geq \frac{31}{32}\) to find appropriate values for \(p\).

The exercise amounts to isolating \(p\) in an inequality, by rearranging terms. We first transform the expression into \(p^5 \leq \frac{1}{32}\), which simplifies the problem into finding the fifth root of both sides.

These steps help identify the interval conditions that \(p\) must meet to satisfy the original condition of the problem, allowing us to determine the correct answer choice.

Complement Probability

Complement probability is a useful technique for finding the likelihood of an event happening at least once. It relies on the logic that an event's probability plus the probability of its complement must equal one.

In our scenario with Bernoulli's trials, instead of calculating the probability of at least one failure directly, we find the complement: the probability of no failures at all. If \(p\) is the success probability in a single trial, the probability of all successes in five trials is \(p^5\).

Thus, the complement probability \(1 - p^5\) gives the likelihood of experiencing at least one failure across trials, aligning with the original exercise's requirement. This approach simplifies problem-solving by handling what's left over, rather than what's wanted, directly.