Cass is taking a 5 question multiple-choice quiz in which each question has 3 choices. He guesses on

Cass is taking a 5 question multiple-choice quiz in which each question has 3 choices. He guesses on all of the questions. What is the probability that Cass answers all of the questions correctly? What is the probability that he answers exactly 3 questions correctly?
bobbi78
Asked May 17, 2010
There's a 1 in 243 chance that he'll get them all right, and a 40 in 243 chance that he'll get 3 right.

The way this is done is we do 3^5 which is 246, and then count up the number of possibilities for each scenario. There is one chance to get all 5 questions right, and then for the wrong answers you have to consider the possibilities of getting the wrong answers wrong in multiple arrangements, such as:
0XX 0XX 0XX 0XX
0XX 0XX 0XX 0XX
0XX 0XX 0XX 0XX
X0X X0X XX0 XX0
X0X XX0 X0X XX0

And then you have to take into account that you can get different questions right, while still only having 3 questions right, such as:
0XX 0XX 0XX
0XX 0XX 0XX
0XX X0X X0X
X0X 0XX X0X
X0X X0X 0XX

When you add it all up, you get 40.
Oni_Kami
Answered May 17, 2010

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