1983 AMC 12 Problem 15

Question

Three balls marked

1,

2

and

3

are placed in an urn. One ball is drawn, its number is recorded, and then the ball is returned to the urn. This process is repeated and then repeated once more, and each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is

6,

what is the probability that the ball numbered

2

was drawn all three times?

\frac1{27}

\frac18

\frac17

\frac16

\frac13

Answer: C

Concepts:conditional probability systematic listing sampling with replacement

Difficulty rating: 1690

Small hint:

List the ordered triples from

\{1,2,3\}

whose sum is

6

Big hint:

The possibilities are the permutations of

(1,2,3)

together with

(2,2,2)

Solution:

The ordered triples with sum

6

are the six permutations of

(1,2,3)

and the triple

(2,2,2).

They are equally likely, and exactly one of the seven has three

2

’s. The conditional probability is therefore

\frac{1}{7}.

Therefore, the correct answer is C.