Multiple Choice Tests

Date: 05/06/97 at 02:41:28
From: nancy
Subject: Probability

I have a question on my math homework that I can't figure out. I think 
it deals with Pascal's triangle, but I just don't understand it.

If you have a test with 6 questions and 4 possible answers to each 
question, and you guess all answers, what is the probability of 
getting all 6 questions right? 

I tried using the 6th row and the 4th column of Pascal's triangle, but 
I don't think that is right.

Can you please help me?

Date: 05/06/97 at 05:53:28
From: Doctor Anthony
Subject: Re: probability

This is relatively easy because you are asked for the probability that 
you get all 6 questions right.

The probability of getting any one question right is 1/4 because there 
are 4 possible answers, only one of which is correct.  

The probability of getting all six questions right is the probability 
of getting one question right times itself six times:

1/4 x 1/4 x 1/4 x 1/4 x 1/4 x 1/4 = (1/4)^6 = 1/4096

Not very good odds!  It's better to study so you don't have to guess!

A more difficult problem would be to figure out the probability of 
getting 3 out of the 6 correct.

Suppose you got the first three correct and the last three wrong.  
We already know the probability of getting one question right: 1/4.  
This means the probability of getting one question wrong must be 3/4 
(you have to either get it right or wrong - there's no middle ground 
here). The probability of this particular sequence of right and wrong 
answers would be:

    1/4 x 1/4 x 1/4 x 3/4 x 3/4 x 3/4  = (1/4)^3 (3/4)^3

However there are many possible sequences for right and wrong answers 
giving 3 correct and 3 incorrect. This is where the terms of Pascal's 
triangle would come in. The number of possible sequences in this case 
is given by: 

            6.5.4
  6_C_3  = -------  = 20
            1.2.3 

So there are 20 possible sequences giving three correct and three 
incorrect answers.

The probability of exactly three correct answers is:

                      20(1/4)^3 (3/4)^3
                   =  540/4096
                   =  135/1024


-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/