Understanding the binomial distribution
Hi, I'm Dr Nic from Creative Maths
including Statistics Learning Centre
This video is about the binomial probability distribution,
where it is applicable and a bit of an idea of where it comes from
This video builds on our previous videos about discrete distributions
and probability distribution models
A binomial distribution is used to model situations
where you have a series of independent
events that all have the same probability and have just two outcomes.
A binomial distribution is defined by the number of successes X
out of a number of trials, n,
all of which have the same probability p.
We state the distribution as
Binomial (n = , p =)
Here is an example.
Luke sells ice creams in his ice cream stand.
He is concerned that his new supply of cones is substandard.
Too many of them are breaking.
From his previous supply he estimates that about 5% of cones break.
This can be expressed as a probability of 0.05.
We define X as being the number of cones that break in a box of 12 cones.
Each cone has a probability of 0.05 of breaking
We can model this as a binomial distribution
where n, the number of trials, equals 12
and p, the probability of success, equals 0.05.
In this case we have defined success as a cone breaking
In statistics, words can mean something different from their everyday meaning.
Luke wants to know the probability that
exactly one cone breaks out of a packet of 12.
He wants to find the probability that x equals one.
We can use a formula, a table, a spreadsheet an app
or a calculator to find the probability.
I particularly like this free app found at online stat book.com.
I enter N of 12, the number of cones in the box,
and a pi of 0.05.
Sometimes the individual probability of success uses the Greek letter pi.
I state that I would like to find out the probability of X being exactly one
and I do this by saying that the probability I'm
interested in is between 1 and 1 inclusive.
You can check on the graph
that you have the probability range that you're interested in.
The answer is 0.3413
and you can see it and read it on the graph
Now say I wanted to know the
probability that two or more cones break in a box of 12
I changed the input to above 1.
By looking at the graph I can see that I've included all values 2 and above.
The probability of two or more cones breaking is 0.1184.
We can use the app to explore further
the likelihood of different numbers of breakages and
different ranges of numbers of breakages.
Understanding the binomial distribution.
We will now look at where this answer comes from.
Now say Luke wanted to know the distribution for one cone.
It would look like this
There are only two values that X can take.
Zero: no breakage or one: one breakage.
There is a 95% probability that the number of breakages is zero
and a 5% probability that the number of breakages is 1.
But what about if there were two cones?
Here is the distribution for binomial N=2, p=0.05.
The most likely thing to happen is that no cones break.
The probability for zero breakages
is the probability that cone one doesn't break which is 0.95 and cone two doesn't
break which is also 0.95.
The combined probability is
0.95 times 0.95 which is 0.9025
The probability of one of the two cones breaking
is the probability of the first one breaking and the second one staying whole
or the first one staying whole and the second one breaking.
That equals 0.05 times 0.95 plus 0.95 times 0.05
which comes 0.095.
Is it possible for both cones to break?
Yes but it's very unlikely
0.05 times 0.05 equals 0.0025.
And here is a distribution of breakages for three cones.
As you can see the likelihood of no breakages is
getting smaller and the spread of numbers is getting greater.
It soon gets too tricky to work out all of the combinations individually
So this is why we use tables or electronic methods.
Requirements for a binomial distribution to be applicable
There are four conditions for a binomial distribution
There are multiple trials.
For each trial there are two possible outcomes.
The probability of success is constant - the same for each trial
and the trials are independent - the outcome of one does not affect the outcome of another.
in our example each of the cones was a trial.
The outcome was either a breakage - success,
or no breakage - failure.
We assume that the rate of 5 percent breakage
stays the same throughout the supply of cones
and we assumed that one cone
breaking is not going to affect whether or not the next cone breaks
That was a quick overview of the binomial distribution
if you would like some more
maybe some examples let me know in the comments below.
Do click on like and subscribe to the channel.
click on the bell to be notified when more videos are
posted and may your probability of success be very close to 1