   SEARCH HOME Math Central Quandaries & Queries  Question from Peter, a student: The numbers 1-400, inclusive, are put into a hat. What is the probability that the first number chosen at random is a multiple of 4 or 17? Express your answer as a common fraction. How do you do this the question without a calculator? Hi Peter,

The probability of a single event, such as choosing a multiple of 4, can be found by the simple probability formula:

# of successful outcomes = 100 since every 4th number is a multiple of 4 = 1/4
# of possible outcomes 400 since there are 400 numbers in the hat

The probability of choosing a multiple of 17 can be found in the same way:

# of successful outcomes = 23 since every 17th number is a multiple of 17 **
# of possible outcomes 400 since there are 400 numbers in the hat

** to find how many multiples of 17 are less than or equal to 400, with or without a calculator, simply divide 400 by 17 to see how many times it goes in, ignoring any remainder you may get.

If we need to want the probability that the number is a multiple of 4 OR a multiple of 17, we must use the OR Probability Formula:

P(A or B) = P(A) + P(B) - P(A and B)

If we let A be the event that we choose a multiple of 4, and B be the event that we choose a multiple of 17, then (A and B) would be the event that the number is a multiple of 4 and a multiple of 17 at the same time. Since 4 and 17 have no common factors (as 17 is prime), all numbers that are multiples of both will be multiples of their product (4*17=68)

There are 5 multiples of 68 less than or equal to 400 (found like ** above for 17).

Therefore, P(A or B) = 100/400 + 23/100 - 5/100 = 118/400 = 59/200 in lowest terms.

Hope this helps,
Leeanne     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.