Consider 5 independent Bernoulli's trials each with probability of success p. If the probability of

Q: Why can't we just calculate the probability of 1 failure, 2 failures, etc., and add them up?

You absolutely could, but it's like trying to walk from Delhi to Mumbai when you could just take a flight! Calculating $P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)$ is tedious and prone to arithmetic errors. The complement rule $1 - P(X=0)$ is the 'flight'—it gets you to the answer in one elegant step.

Q: Wait, why is the probability of no failure $p^5$?

This is the beauty of independence! In Bernoulli trials, the outcome of one trial doesn't affect the others. So, if the probability of success in one trial is $p$, the probability of success in five consecutive trials is $p \times p \times p \times p \times p = p^5$. It's just simple multiplication of independent events.

Q: Why do we need to include $0 \le p$ in the final answer?

This is a classic JEE trap! Algebraically, you might solve $p^5 \le \frac{1}{32}$ and get $p \le \frac{1}{2}$. But in the physical world of probability, $p$ is a measure of likelihood. A probability can never be negative. Always remember the hidden constraint $0 \le p \le 1$ whenever you solve for a probability variable.

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

Options:

(A) $(3/4, 11/12]$

(B) $[0, 1/2]$

(C) $(11/12, 1]$

(D) $[1/2, 3/4]$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before a series of five independent Bernoulli trials. Each trial is a moment of truth, a coin flip of destiny where the probability of success is $p$.

The problem asks us to analyze the condition where the probability of at least one failure is at least $\frac{31}{32}$.

At first glance, this might seem like a daunting task. You might be tempted to calculate the probability of exactly one failure, then two, then three, four, and five, and sum them all up.

But stop! In the world of JEE Advanced, efficiency is just as important as accuracy. Whenever you encounter the phrase 'at least one', your mathematical intuition should immediately pivot to the **complement rule**.

The event 'at least one failure' is the exact opposite of 'zero failures'. By calculating the probability of the complement, we simplify our lives immensely.

### The Power of Independence

What does 'no failure' actually mean? It means that every single one of our five trials resulted in a success.

Because these trials are independent, the outcome of one does not influence the others. This allows us to simply multiply the probabilities.

If the probability of success in one trial is $p$, then the probability of five consecutive successes is $p 	imes p 	imes p 	imes p 	imes p$, which is $p^5$. This is the probability of the complement event.

Now, we can write our condition as:

$$1 - p^5 \ge \frac{31}{32}$$

This is the heart of the problem.

### The Algebraic Dance

Now, let's solve this inequality with precision. We want to isolate $p$.

First, rearrange the terms:

$$1 - \frac{31}{32} \ge p^5$$

Simplifying the left side gives us:

$$\frac{32 - 31}{32} = \frac{1}{32}$$

So, we have $\frac{1}{32} \ge p^5$, or $p^5 \le \frac{1}{32}$.

This is where the magic happens. We recognize that $\frac{1}{32}$ is simply $\left(\frac{1}{2}ight)^5$.

Thus, we have:

$$p^5 \le \left(\frac{1}{2}ight)^5$$

Taking the fifth root of both sides, we arrive at $p \le \frac{1}{2}$.

### The Final Constraint

We are almost at the finish line, but we must not forget the physical reality of probability. A probability $p$ is not just any variable; it is a value that must exist between $0$ and $1$.

Our algebraic solution $p \le \frac{1}{2}$ must be bounded by this fundamental constraint.

Therefore, we combine $p \le \frac{1}{2}$ with $p \ge 0$ to get the final interval:

**$$p \in \left[0, \frac{1}{2}ight]$$**

You have successfully navigated the trap, used the complement rule, and respected the bounds of probability. This is the kind of rigorous thinking that defines a true JEE scholar.

Animated Solution for Mathematics - Probability: 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

Select Answer:

Visualized Solution

Defining the Trials

The Given Condition

The Complement Rule

Probability of No Failure

Calculating $P (no failure)$

Setting up the Inequality

Rearranging the Terms

Simplifying the Inequality

Taking the Fifth Root

The Probability Constraint

Final Interval

The Sigma Insight: Random Variables and Probability Distributions

Analyzing the Setup

The Power of Independence

The Algebraic Dance

The Final Constraint

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