Calculate Scaled Probability of Minimum Selected Number

To determine the value of $\frac{125}{4} p_2$, we proceed step-by-step:

Step 1: Define the Sample Space and Event

The sample space is $S = \{1, 2, 3, \dots, 100\}$, which contains $100$ elements. Three numbers are chosen sequentially with replacement from $S$.

Let $p_2$ denote the probability that the minimum of the three chosen numbers is at most $40$.

Step 2: Use the Complementary Event

The complement of the event "the minimum is at most $40$" is the event "the minimum is strictly greater than $40$".

For the minimum of the three chosen numbers to be strictly greater than $40$, each of the three numbers must be selected from the set: $S' = \{41, 42, 43, \dots, 100\}$

The number of favorable choices in $S'$ is: $|S'| = 100 - 40 = 60$

Step 3: Calculate the Complementary Probability

Since the numbers are drawn with replacement, the draws are independent. The probability that a single chosen number is strictly greater than $40$ is: $P(\text{Number} > 40) = \frac{60}{100} = \frac{3}{5}$

Thus, the probability that all three chosen numbers are greater than $40$ (i.e., $\text{minimum} > 40$) is: $P(\text{minimum} > 40) = \left(\frac{3}{5}\right)^3 = \frac{27}{125}$

Step 4: Calculate $p_2$

Using the complement rule: $p_2 = 1 - P(\text{minimum} > 40) = 1 - \frac{27}{125} = \frac{98}{125}$

Step 5: Compute the Required Value

Now, we substitute $p_2$ into the expression $\frac{125}{4} p_2$: $\frac{125}{4} p_2 = \frac{125}{4} \times \frac{98}{125} = \frac{98}{4} = 24.5$

Thus, the value of $\frac{125}{4} p_2$ is $24.5$.