Difference between Bernouilli Probability and Binomial Probability

[eloisegrace]

Hello fellow ANer's!

I was wondering if someone could give me a little bit of a description of the difference between Bernouilli and Binomial Probability? Is Bernouilli for one trial where as Binomial is for multiple?

Both distributions are on the formula sheet but I have heard that really only Binomial Probability comes up in exams?

Thanks!
Eloise

P.S. How likely are either of these to come up on exam one (non CAS) due to the heavy calculator component?

[james.358]

Hey Eloise!

The Bernoulli Distribution is a special case of discrete probability distribution where there is only one interested outcome (i.e. success or failure). It doesn't appear much in exams as there is hardly, if any calculations involved.

You can essentially think of binomial distribution as a method of calculating the probability of multiple trials of Bernoulli events.

Hope this helps!
James

[keltingmeith]

Side-note: The definition of the binomial distribution is as the sum of n iid (independent and identically distributed) Bernoulli random variables. That is, if has distribution Bernoulli(p) (where p is the probability of success), then:



has distribution Binomial(n,p), where n is the number of trials and p is the probability of success.


In the less mathematical sense, it's what james.lhr has said - the binomial distribution is just the same thing as calculating a bunch of multiple trials of the Bernoulli distribution, though the idea that each separate event is independent is super important (though often missed in VCE methods and taken for granted, which often then leads to confusion when you learn some stuff in specialist where independency and dependency can lead to using different equations, and people don't know which to use).


(also sorry james for interrupting an already well-answered question, but things like this are often lost in probability teaching and are some of the reasons I love prob and stats so much )

[eloisegrace]

james.lhr: Thankyou for that simple and informative answer! That has fixed up all confusion I had with this . I wish you luck for methods this year and for VCE in general (fellow class of 21er!)

keltingmeith: I appreciate the time you have spent to fully explain this Even though I am not taking specialist I will definitely use the fact that they are independent in my methods studies!

[peterdong]

A Bernoulli random variable X is a random variable that satisfies P(X=1)=p, P(X=0)=1−p. A canonical example is a coin flip which has p=1/2. In fact, you can think of a Bernoulli random variable is just a weighted coin, which comes up 1 with some probability and 0 otherwise. A binomial random variable with parameters n,p is what you get when you count the number of 1's (successes) that come up in a string of n independent Bernoulli random variables, each with parameter p. Another way to say this is that a binomial random variable is the sum of independent and identically distributed Bernoulli random variables.