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### Binomial Probability

Date: 04/14/99 at 13:38:21
From: Amy Pacyon
Subject: Probability

I am unfamiliar with the binomial probability formula; it was not
covered in class. The question is:

A 10-question multiple choice exam is given, and each question has
five possible answers. Pasxal takes this exam and guesses at every
question. Use the binomial probability formula to find the probability
(to 5 decimal places) that

A) he gets exactly 2 questions correct;
B) he gets no questions correct;
C) he gets at least one question correct (use the information from
part B to answer this part);
D) he gets at least 9 questions correct;
E) without using the binomial probability formula, determine the
probability that he gets exactly 2 questions correct;
F) Compare your answers to parts a and f.  If they are not the same
explain why.

Date: 04/14/99 at 15:03:23
From: Doctor Anthony
Subject: Re: Probability

You say that you have not yet covered binomial probability, so I will

Flipping coins is an example of binomial probability, where there are
repeated trials, with constant probability of success (say getting a
head) at each trial. As an example suppose there are 4 with
probability (1/2) of success at each trial.

In the general binomial model if we carry out n trials, with
probability p of success at each trial and probability q of failure,
where p+q = 1, then if we want a probability of say r successes, one
possible sequence is r successes followed by n-r failures.

The probability of this sequence is  ppppp to r terms x qqqqq to n-r
terms.

However, we can get r successes in a whole range of sequences, the
number of sequences being the same as the number of possible sequences
of r p's and n-r q's.

n!
The number of sequences  =  --------      = C(n,r)
r! (n-r)!

So the total probability of r successes and n-r failures is

P(r) =  C(n,r) p^r q^(n-r)

Returning to our problem with 4 coins, we have n = 4, p = 1/2, and
q = 1/2, and we require a probability of exactly 2 heads.  So r = 2.

P(2 heads) = C(4,2) (1/2)^2 (1/2)^2

=  6 x 1/4 x 1/4

=   6/16

=  3/8

Now we look at the question of the multiple choice exam.  I will work
through part (A) and leave you to try and complete the question.

>A) he gets exactly 2 questions correct.

In this example n = 10,  p = 1/5,  q = 4/5,  r = 2

P(2) = C(10,2) x (1/5)^2 x (4/5)^8  =  0.3019899

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/

Associated Topics:
High School Permutations and Combinations
High School Probability

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