2006 AMC 10B Problems/Problem 17

Problem

Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process the contents of the two bags are the same?

Solution

After Alice puts the ball into Bob's bag, his bag will contain six balls: two of one color and one of each of the other colors. After Bob selects a ball and places it into Alice's bag, the two bags will have the same contents if and only if Bob picked one of the two balls in his bag that are the same color. Because there are six balls in the bag when Bob makes his selection, the probability of selecting one of the same colored pair is . -Clara Garza

Video Solution

https://youtu.be/5UojVH4Cqqs?t=1160

~ pi_is_3.14

Solution

Since there are the same amount of total balls in Alice's bag as in Bob's bag, and there is an equal chance of each ball being selected, the color of the ball that Alice puts in Bob's bag doesn't matter. Without loss of generality, let the ball Alice puts in Bob's bag be red.

For both bags to have the same contents, Bob must select one of the red balls out of the balls in his bag.

So the desired probability is .

Solution 2

Suppose Alice selects the blue ball. This happens with chance.

Then Bob has to select blue, too and now he has blue balls and total balls.

This has a chance.

Since Alice can choose total colours and the probability for each is the same since there are the same number of balls for each, the probability is

~mathboy282

See Also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.