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?

Question

Gabriel Allred · Accepted Answer

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

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} #

Generally speaking, the probability of choosing a number that is divisible by 6 is ~1/6 if you have n slips of paper numbered 1 to N (where N is a large positive integer, say 100). If N is exactly divisible by 6, 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 had selected 24 instead, there would have been 4: 61, 62, 63, 64 = 6,12, 18, and 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}#