Question

What is the probability that it is a multiple of 5, which is less than 200 ?

Hint:

To count multiples of a number 'k' in a range [a, b], you can calculate (b)/(k) - (a-1)/(k). Here, for multiples of 5 in [100, 199], it would be (199)/(5) - (99)/(5) = 39 - 19 = 20. This is a very fast method for large ranges.

Answer 1

We need to find the probability that a randomly chosen three-digit number is a multiple of 5 and is also less than 200.

The probability of an event is calculated as:
P(Event) = Number of favorable outcomesTotal number of possible outcomes Total number of possible outcomes:
As calculated in the previous part, the total number of three-digit numbers is 900.

Number of favorable outcomes:
We need to count the three-digit numbers that are multiples of 5 and are less than 200.
The three-digit numbers start from 100. So we are looking for multiples of 5 in the range [100, 199].
The multiples of 5 in this range are: 100, 105, 110, ..., 195.
This forms an arithmetic progression with first term a = 100, last term aₙ = 195, and common difference d = 5.
We can find the number of terms (n) using the formula aₙ = a + (n-1)d.
195 = 100 + (n-1)5 95 = (n-1)5 n-1 = (95)/(5) = 19 n = 19 + 1 = 20 So, there are 20 favorable outcomes.

Calculating the probability:
P(multiple of 5 and < 200) = Favorable outcomesTotal outcomes = (20)/(900) Simplifying the fraction:
P(multiple of 5 and < 200) = (2)/(90) = (1)/(45) The probability is (1)/(45).