1988 AMC 12 Problem 12

Question

Each integer

1

through

9

is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. Which digit is most likely to be the units digit of the sum of Jack’s integer and Jill’s integer?

0

1

8

9

each digit is equally likely

Answer: A

Concepts:probability modular arithmetic counting ordered pairs

Difficulty rating: 1290

Small hint:

There are

81

equally likely ordered pairs

Big hint:

Count pairs whose sum has each residue modulo

10

Solution:

For a units digit of

0,

the ordered pairs are

(1,9),(2,8),\ldots,(9,1),

giving

9

pairs. Each other units digit occurs

8

times among the

81

ordered pairs. Therefore

0

is most likely.

Thus the correct answer is A.