On a game show, a contestant is given three keys, each of which opens exactly one of three identical...
On a game show, a contestant is given three keys, each of which opens exactly one of three identical boxes. The contestant assigns each key to one of the boxes and wins the amount of money contained in any box that is opened by the key assigned to it. The first box contains $1.5, the second $150, and the third $1500. What is the probability that a contestant will win more than $1500?
Answer/Solution
1/6
Steps/Work
Let's call the boxes that contain $1.5, $150, and $1500, respectively, Box A, Box B, Box C. These are opened, respectively, by Key A, Key B, and Key C.
We want to know the probability of winning more than $1500. Notice that if the distribution of keys is:
Box A = Key B
Box B = Key A
Box C = Key C
then the contestant wins exactly $1500, not more than $1500. The only configuration that leads to winning more than $1000 is:
Box A = Key A
Box B = Key B
Box C = Key C
i.e., getting all three keys correct. That's the only way to be more than $1500. So, really, the question can be rephrased: what is the probability of guessing the order of keys so that each key matches the correct box?
Well, for a set of three items, the number of possible permutations is 3! = 3*2*1 = 6.
Of those 6 possible permutations, only one of them leads to all three keys being paired with the right box. So, the answer is
Probability = 1/6
Answer = C
We want to know the probability of winning more than $1500. Notice that if the distribution of keys is:
Box A = Key B
Box B = Key A
Box C = Key C
then the contestant wins exactly $1500, not more than $1500. The only configuration that leads to winning more than $1000 is:
Box A = Key A
Box B = Key B
Box C = Key C
i.e., getting all three keys correct. That's the only way to be more than $1500. So, really, the question can be rephrased: what is the probability of guessing the order of keys so that each key matches the correct box?
Well, for a set of three items, the number of possible permutations is 3! = 3*2*1 = 6.
Of those 6 possible permutations, only one of them leads to all three keys being paired with the right box. So, the answer is
Probability = 1/6
Answer = C