Question

You have the numbers 1-24 written on a slip of paper. If you chose one slip at random what is the probability that you will not select a number which is divisible by 6?

Answer 1

The probability is `\frac{5}{6}`

Explanation:

Let A be the event of selecting a number divisible by 6 and B be the event of selecting a number not divisible by 6:
`P(A) = \frac{1}{6}`
`P(B) = P(not A) = 1 - P(A)`
`= 1-\frac{1}{6} = \frac{5}{6}`

In general, if you have n slips of paper numbered 1 to N (where N is a big positive integer say 100) the probability of selecting a number divisible by 6 is ~1/6 and if N is exactly divisible by 6, then the probability is exactly 1/6
i.e.

`P(A) = \frac{1}{6} iff N equiv 0 mod 6`

if N is not divisible exactly by 6 then you would calculate the remainder, for example if N = 45:
`45 equiv 3 mod 6`
(6*7 = 42, 45-42 = 3, the remainder is 3)

The greatest number less than N that is divisible by 6 is 42,
and `because \frac{42}{6} = 7` there are 7 numbers divisible between 1 to 45
and they would be `6*1,6*2, ... 6*7`

if you instead chose 24 there would be 4: and they would be 61,62, 63,64 = 6,12,18,24

Thus the probability of choosing a number divisible by 6 between 1 and 45 is `\frac{7}{45}` and for 1 to 24 this would be `\frac{4}{24} = \frac{1}{6}`

and the probability of choosing a number not divisible by 6 would be the complement of that which is given by `1 - P(A)`
For 1 to 45 it would be: `1 - \frac{7}{45} = \frac{38}{45}`
For 1 to 24 it would be: `1 - \frac{1}{6} = \frac{5}{6}`