What is the probability that a randomly selected three-digit number has the property that one digit is equal to the product of the other two? Express your answer as a common fraction.

A naturalist leads whale watch trips every morning in March. The number of whales seen has a Poisson distribution with a mean of 2.1 Find the probability that on a randomly selected​ trip, the number of whales seen is 3. Round to four decimal places.

13/255

To find the probability that a randomly selected three-digit number has the property described, we need to determine the number of favorable outcomes (numbers with the desired property) and the total number of possible outcomes (all three-digit numbers).

Let's consider the possible values for the three digits of the number: the hundreds digit (H), the tens digit (T), and the units digit (U).

If one digit is equal to the product of the other two digits, we have the following possibilities:

Case 1: H = T * U
Case 2: T = H * U
Case 3: U = H * T

We'll calculate the number of favorable outcomes for each case:

Case 1: For H = T * U, we know that the possible values for T and U are from 1 to 9, since 0 cannot be a leading digit. So, for each value of H (1-9), we have 9 * 9 = 81 favorable possibilities.

Case 2: For T = H * U, we again have the possible values for H and U ranging from 1 to 9. For each value of T (1-9), we have 9 * 9 = 81 favorable possibilities.

Case 3: For U = H * T, the possible values for H and T range from 1 to 9. For each value of U (1-9), we have 9 * 9 = 81 favorable possibilities.

Therefore, the total number of favorable outcomes is 81 + 81 + 81 = 243.

Now let's determine the total number of three-digit numbers:

Since the hundreds digit cannot be zero (leading zero), there are 9 choices for the hundreds digit (1-9). For the tens and units digits, each can have any value from 0 to 9, so there are 10 choices for each.

Therefore, the total number of three-digit numbers is 9 * 10 * 10 = 900.

To find the probability, we divide the number of favorable outcomes by the total number of possible outcomes:

Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 243 / 900

To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 27:

Probability = 9 / 36

Further simplifying the fraction, we can divide both the numerator and denominator by 9:

Probability = 1 / 4

Therefore, the probability that a randomly selected three-digit number has the property that one digit is equal to the product of the other two is 1/4.