Problem 21 - A Complete Resource Book in Mathematics for JEE Main

Question

Numbers are selected at random, one at a time, from the two-digit numbers \(00,01,02, \ldots, 99\) with replacement. An even E occurs if and only if the product of the two digits of selected number is 18 . If four numbers are selected, the probability that the event E occurs at least 3 times, is (A) \(\frac{99}{(25)^{4}}\) (B) \(\frac{86}{(25)^{4}}\) (C) \(\frac{74}{(25)^{4}}\) (D) \(\frac{97}{(25)^{4}}\)

Step-by-Step Solution

Answer

The probability is \(\frac{97}{(25)^4}\), option (D).

Step 1: Identify Two-Digit Numbers

The two-digit numbers range from 00 to 99, giving us 100 possible choices. A particular event E will occur if the product of the two digits is 18.

Step 2: Determine Numbers with Product 18

Calculate which pairs of digits from 0 to 9 have a product of 18: \((2, 9), (3, 6), (6, 3), (9, 2)\). These are valid digit pairs that satisfy the condition. The corresponding numbers are 29, 36, 63, and 92, with four possibilities in total.

Step 3: Calculate Probability of Event E

The probability of selecting a number such that the event E occurs is the number of successful outcomes divided by total possible outcomes: \[ P(E) = \frac{4}{100} = \frac{1}{25} \].

Step 4: Define Binomial Probability for Event Occurrences

Let X be the random variable representing the number of times event E occurs. We want to find the probability of E occurring at least 3 times when 4 numbers are selected. This can be modeled using a binomial distribution with parameters \( n = 4 \) and \( p = \frac{1}{25} \).

Step 5: Calculate Probability of E Occurring 3 Times

Use the binomial probability formula: \[ P(X=3) = \binom{4}{3} \left( \frac{1}{25} \right)^3 \left( \frac{24}{25} \right)^1 \].Substitute values to get: \[ P(X=3) = 4 \times \frac{1}{15625} \times \frac{24}{25} = \frac{96}{(25)^4} \].

Step 6: Calculate Probability of E Occurring 4 Times

Use the binomial probability formula: \[ P(X=4) = \binom{4}{4} \left( \frac{1}{25} \right)^4 \].There is only one way this can occur: \[ P(X=4) = \frac{1}{(25)^4} \].

Step 7: Sum Probabilities for At Least 3 Occurrences

The probability that event E occurs at least 3 times is given by: \[ P(X \geq 3) = P(X=3) + P(X=4) = \frac{96}{(25)^4} + \frac{1}{(25)^4} = \frac{97}{(25)^4} \].

Key Concepts

Binomial Distribution

The binomial distribution is a probability distribution that summarizes the likelihood of a value taking on one of two independent states. These states are traditionally labeled "success" or "failure." In this context, we are interested in how many times an event, labeled as a success, happens in a fixed number of trials.When using a binomial distribution to model probabilities, several conditions must be met:

• The number of trials, denoted as \( n \), is fixed.
• Each trial is independent of the others.
• Each trial can result in just two possible outcomes: success or failure.
• The probability of success, \( p \), is the same for each trial.

In our exercise, selecting four numbers represents our trials, and we are interested in the event "E occurs," meaning the product of the digits of the number equals 18.Knowing the binomial distribution helps us predict the number of times this event happens within four tries.

Two-Digit Numbers

Two-digit numbers are numbers ranging from 00 to 99. They play a critical role in this exercise since we are drawing from this specific set. There are a total of 100 numbers in this range. Each number can be seen as composed of two separate digits, ranging from 0 to 9. When we talk about two-digit numbers in this exercise, we're specifically interested in those where the product of the digits equals 18. This subset is crucial because we're identifying our successful outcomes based on these specific numbers.

Successful Outcomes

In probability, successful outcomes refer to the specific cases we are interested in observing. They represent what we define as "success" in a given scenario. For this exercise, a successful outcome is selecting a number where the product of the two digits is 18. Identifying these outcomes involves:

• Determining the pairs of digits that multiply to give 18: (2, 9), (3, 6), (6, 3), and (9, 2).
• Finding the corresponding two-digit numbers: 29, 36, 63, and 92.

These four numbers are our successes. They form the basis for calculating probabilities and applying the binomial distribution.

Binomial Probability Formula

The binomial probability formula helps calculate the likelihood of a given number of successes in a fixed number of trials. The formula is expressed as:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]where:

• \( P(X = k) \) is the probability of getting exactly \( k \) successes in \( n \) trials.
• \( \binom{n}{k} \) represents the number of ways to choose \( k \) successes from \( n \) trials (known as the binomial coefficient).
• \( p \) is the probability of success on a single trial.
• \( (1-p)^{n-k} \) represents the probability of the remaining trials resulting in failures.

In our example, we apply this formula to calculate the probabilities of 3 and 4 successful outcomes when 4 numbers are drawn, using \( p = \frac{1}{25} \), derived from the probability of drawing each successful number.