Question 971076: what is the probability that a number selected at random from the first 500 positive integers is exactly divisible by 3 or 4
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
The answer is simply the number positive integers between 1 and 500 inclusive that are divisible by either 3 or 4 divided by 500. Note that we start with 1 rather than zero because the problem specifies 'positive' integers rather than non-negative integers.
We start by calculating the number of integers from 1 to 500 that are divisible by 3. There are 500 numbers and every third one is divisible by 3, so divide by 3 and throw away the decimal part. 166.
Then we need the number of integers from 1 to 500 that are divisible by 4. Again, we have 500 numbers and every fourth one is divisible by 4. So 500 divided by 4 is 125 with no decimal part to throw away.
But if you just add those two numbers, the numerator of your probability fraction is going to be too large. That is because of the 166 numbers in the given range that are divisible by 3, there are some that are also divisible by 4. Likewise, some of the divisible by 4 numbers are also divisible by 3. These numbers are the ones that are in the range and divisible by 12, and if you just add the 3s and the 4s, you have counted the 12s twice.
500 divided by 12 is 41 (toss the decimal), and now we can calculate the numerator of the fraction: 166 + 125 - 41 = 250
Hence \ \large\wedge\LARGE x\ \in\ \mathbb{Z}^+, x\ \leq\ 500\right)\ =\ 0.5)
John
My calculator said it, I believe it, that settles it
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