14th Amdt

19th Amdt
















See also



Multiple Choice Test: Binomial Probability

Date: 08/05/97 at 18:55:12
From: Heather
Subject: Multiple choice test

A multiple choice test consists of 9 questions with 5 choices for each
answer. If a student guesses randomly, find the probability of each of 
the following events:

  1. The student gets 5 correct
  2. The student gets at least 5 correct
  3. The student gets 2 correct answers
  4. The student gets fewer than 3 correct answers.

Date: 08/06/97 at 11:55:26
From: Doctor Anthony
Subject: Re: Multiple choice test

This is binomial probability with  n = 9,  p = 1/5,  q = 4/5

  1. P(5) = 9C5 (1/5)^5 (4/5)^4 = 0.016515

  2. At least 5 is = P(5) + P(6) + P(7) + P(8) + P(9)

     P(6) = 9C6 (1/5)^6 (4/5)^3 = .0027525   and so on.....

  3. P(2) = 9C2 (1/5)^2 (4/5)^7 = 0.301989

  4. P(< 2) = P(0) + P(1) 

       P(0) =  (4/5)^9 = 0.1342177

       P(1) = 9C1 (1/5) (4/5)^8  = 0.301989

So   P(<2) = Sum of these = 0.43621 

-Doctor Anthony,  The Math Forum
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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.

Thank you in advance for your help.

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 
start with a general note on the topic.

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 

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.

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   




Probability of Collision


Probability of Collision Expert Driver Novice Driver Total Ratio
Billion miles driven 1,056 455 1,511 .43
collisions 24,087 17,729 41,816 .74
collisions per billion miles 22.8 39 27.7 1.71
Percent of miles driven 70% 30% 100% .43
Percent of collisions 57.6% 42.4% 100% .74


The 71% higher probability per mile that Novice Drivers will have an automobile collision contributes to an increase in the collision rate for Expert Drivers.  To determine exactly how much higher the collision rate for Expert Drivers is because of Novice Drivers it's necessary to calculate the rate per one million miles that both Expert Drivers and Novice Drivers are expected to have an collision.  Let Ne be the number of collisions per million miles that an Expert Drivers is expected have a single driver auto collision, and Nn that a Novice Driver will, and then there are two equations and two variables.  The total number of collisions per million miles that an Expert Driver is expected to have an collision, Re, is the sum of his likelihood per million miles of having a single driver collision Ne, the square of this probability to represent a two driver collision involving another Expert Driver Ne2, and Ne  times Nn  to represent a two driver collision involving a Novice Driver.  For simplicity, collisions involving more than two drivers are omitted, but they are rare enough that the ratios below won't change significantly and it's unlikely that the probability of either driver to have a multiple car crash is much different than the probability of a two driver collision:

(Ne +  .7Ne2  +.3 NeNn) x 1,056 billion miles driven = 24,087 collisions

Re = Ne +  .7Ne2  + .3NeNn = 22.8

The equation for Novice Drivers is similar:

(Nn +  .3Nn2  + .7NeNn) x 455 billion miles driven = 17,729 collisions

Rn = Nn +  Nn2  + NeNn = 39



jewn McCain

ASSASSIN of JFK, Patton, many other Whites

killed 264 MILLION Christians in WWII

killed 64 million Christians in Russia

left 350 firemen behind to die in WTC

holocaust denier extraordinaire--denying the Armenian holocaust

millions dead in the Middle East

tens of millions of dead Christians

LOST $1.2 TRILLION in Pentagon
spearheaded torture & sodomy of all non-jews
millions dead in Iraq

42 dead, mass murderer Goldman LOVED by jews

serial killer of 13 Christians

the REAL terrorists--not a single one is an Arab

serial killers are all jews

framed Christians for anti-semitism, got caught

legally insane debarred lawyer CENSORED free speech

mother of all fnazis, certified mentally ill

10,000 Whites DEAD from one jew LIE

moser HATED by jews: he followed the law Jesus--from a "news" person!!

1000 fold the child of perdition


Modified Tuesday, November 17, 2009

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