Numbers are selected at random, one at a time, from the two-digit numbers 00, 01, 02, ..., 99 with r

Q: Wait, why is the total number of outcomes 100? Shouldn't it be 99 because we start from 01?

That is a very common trap! The problem explicitly states the set includes '00, 01, 02, ..., 99'. If you count from 00 to 99, there are exactly 100 numbers. Always read the range carefully—it is the foundation of your probability calculation.

Q: Why can't we just use simple counting instead of the Binomial Distribution?

You absolutely could use counting, but it would be tedious! The Binomial Distribution is a powerful tool designed specifically for independent trials with two outcomes (success or failure). Since we are selecting with replacement, each of the 4 trials is independent, making the Binomial formula the most elegant and efficient path to the solution.

Q: Why do we calculate $P(X=3) + P(X=4)$? What does 'at least 3' mean?

Great question! 'At least 3' means the event must occur 3 times OR 4 times. Since these are mutually exclusive scenarios (you cannot have exactly 3 successes and exactly 4 successes simultaneously), we simply add their individual probabilities together to get the total probability for the condition $X \ge 3$.

Question

Numbers are selected at random, one at a time, from the two-digit numbers 00, 01, 02, ..., 99 with replacement. An event $E$ occurs if only if the product of the two digits of a selected number is 18. If four numbers are selected, find probability that the event $E$ occurs at least 3 times.

Sigma Percentile · Accepted Answer

### The Geometry of Chance: A Journey into Probability

Welcome, fellow traveler, to the fascinating world of probability. Today, we are not just solving a problem; we are dissecting the very nature of randomness.

We are looking at a scenario where we select four numbers from the set $\{00, 01, 02, \dots, 99\}$ with replacement. This "with replacement" clause is our best friend—it ensures that every single draw is independent, keeping our probability constant.

### Mapping the Sample Space

First, we must define our universe. We are picking from the set $\{00, 01, 02, \dots, 99\}$.

A common mistake is to assume there are 99 numbers, but counting from 00 to 99 gives us exactly $n(S) = 100$ distinct outcomes. Because we are selecting with replacement, the total number of outcomes remains 100 for every draw.

### The Target Event

Now, let us define our target event $E$: the product of the digits of the selected number must be 18. We need to find pairs of digits $(d_1, d_2)$ such that $d_1 	imes d_2 = 18$.

The pairs are $(2, 9), (9, 2), (3, 6),$ and $(6, 3)$. Since these are two-digit numbers, the order matters, giving us the favorable set $\{29, 92, 36, 63\}$.

There are exactly 4 favorable outcomes. Thus, the probability of success in a single trial is:

$$p = \frac{4}{100} = \frac{1}{25}$$

Consequently, the probability of failure is:

$$q = 1 - p = \frac{24}{25}$$

### The Binomial Engine

We are performing $n = 4$ trials. Since each trial is independent and has a constant probability of success, we are in the realm of the **Binomial Distribution**.

We want the event to occur "at least 3 times." This means we are looking for the sum of the probabilities of exactly 3 successes and exactly 4 successes:

$$P(X \ge 3) = P(X=3) + P(X=4)$$

The general formula for this is:

$$P(X=r) = \binom{n}{r} p^r q^{n-r}$$

### The Final Calculation

Let us compute these probabilities. For $r=3$:

$$P(X=3) = \binom{4}{3} \left(\frac{1}{25}ight)^3 \left(\frac{24}{25}ight)^1 = 4 	imes \frac{24}{25^4} = \frac{96}{25^4}$$

For $r=4$:

$$P(X=4) = \binom{4}{4} \left(\frac{1}{25}ight)^4 \left(\frac{24}{25}ight)^0 = 1 	imes \frac{1}{25^4} = \frac{1}{25^4}$$

Adding these together, we get the final result:

$$P(X \ge 3) = \frac{96}{25^4} + \frac{1}{25^4} = \mathbf{\frac{97}{390625}}$$

### Conclusion

Through the lens of the Binomial Distribution, a complex problem of repeated selections collapses into a simple, elegant fraction. You have successfully navigated the sample space, identified the favorable outcomes, and applied the binomial theorem to reach the truth.

Visualized Solution

Understanding the Sample Space

Defining Event $E$ : Product of Digits

Calculating Probability of Success $p$

Identifying the Binomial Distribution

The Binomial Formula

Calculating $P (X = 3)$

Calculating $P (X = 4)$

Final Summation

The Sigma Insight: Binomial Distribution

The Geometry of Chance

Mapping the Sample Space

The Target Event

The Binomial Engine

The Final Calculation

Conclusion

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