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The Binomial Distribution

Posted by Diego em Janeiro 15, 2015


 

In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p (source: wikipedia)

The dbinom function calculates the “probability of success” on a given experiment.

 

There are 3 main parameters, the number of successes for which you want to know the probability; the size (number of trials) and the probability of success on each trial.

 

So, for example, if we want to know the probability of success on the toss of a coin (let’s say “heads” mean success), we can do

dbinom(1, size = 1, prob = 1/2)

[1] 0.5


Clearly, the probability of “failure” (tails) is the same:

dbinom(0, size = 1, prob = 1/2)

[1] 0.5

 

 

If we want to know the probability of 2 heads on 2 tosses of a coin:

dbinom(2, size = 2, prob = 1/2)

[1] 0.25

 

The same way, one success can be calculated by

dbinom(1, size = 2, prob = 1/2)

0.5

Resulting 50%, which makes sense because when flipping a coin the possibilities are HH, HT, TH, TT and we can easily see that on 50% of the cases we have 1 success.

 

Second example, lets says you have a test to take with 5 multiple choice questions, each question has 5 alternatives.

What is the probability of getting exactly 3 questions right just answering them at random.

 

dbinom(3, size=5, prob=1/5)

[1] 0.0512

 

Its 5% so you better study!

 

 

It’s easier to verify this formula if we think as 1 hit in 2 questions with 3 alternatives each. So let’s say that the correct answer is C than B, we can guess:

AA
AB
AC
BA
BB
BC
CA
CB – we don’t want this because we want exactly 1 hit
CC

 

4 out of 9 = 4/9 = 0.44

dbinom(1, size=2, prob=1/3)

0.4444444

 

 

Back to the questions, we can see the probability of getting 0 to 5 questions right

 

x = seq(0,5,by=1)

probx = dbinom(x, size=5, prob = 1/5)

cbind(x, probx)

 

     x   probx

[1,] 0 0.32768

[2,] 1 0.40960

[3,] 2 0.20480

[4,] 3 0.05120

[5,] 4 0.00640

[6,] 5 0.00032

plot(x, probx, type = ‘h’)

image

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