Question

A number is selected at random from the first 1000 natural numbers. What is the probability that the number will be a multiple of 4 or a perfect square?

Answer 1

A number is selected at random from the first 1000 natural numbers. What is the probability that the number will be a multiple of 4 or a perfect square is 0.266

Step-by-step explanation:

Here total number is 1000.

There are 250 numbers between 1 and 1000 which is a multiple of 4 and are given as follows.

4, 8, 12, ............., 992, 996,1000.

Then by using the formula for number of terms in an arithmetic sequence

That is $n=\frac{Last~ term-first ~term}{common~difference} +1\\n=\frac{1000-4}{4}+1\\n=249+1=250$

Also there are 31 perfect squares between 1 and 1000 and are given below.

1, 4, 9, 16, 25, ,,,,,,,,,,,,,,,,,,,,,, 900, 961.

Of these perfect squares 4, 16, 36, 64, 100, 144, 196, 256, 324, 400, 484, 576, 676, 784, 900 are also the multiples of 4.

Therefore the numbers which are multiple of 4 or a perfect square are 250+31-15=266.

Therefore the probability that the number will be a multiple of 4 or a perfect square is $\frac{266}{1000}$=0.266