Introduction to Bernoulli Trials Video Lecture | Mathematics (Maths) Class 12 - JEE

Let's take a look at the Bernoulli distribution,
another important discrete probability distribution.
Here's an example.
Suppose we toss a fair coin once.
What is the distribution of the number
of heads in that single toss?
Well it's going to have a Bernoulli distribution.
Suppose we have a single trial,
where each trial can result in one of
two possible mutually exclusive outcomes.
And we're going to label those two outcomes
success and failure.
The probability of success on that trial
is going to be represented by little p.
And the probability of failure must be 1-p.
We're going to let the random variable X
equal 1 if a success occurs, and 0 if
a failure occurs.
And if these conditions hold, then X has
what we call the Bernoulli distribution,
where the probability the random
variable X takes on the value little x
is equal to p^x [times] (1-p)^(1-x).
And we sometimes write this as just
p(x), you might see it written that way sometimes.
Now what we should do is say what values
X can take on,
and so we write here: for little x being equal to 0 or 1.
And so all of this we refer to sometimes
as the probability mass function of the Bernoulli distribution.
And if we were to write this out for the individual values,
we'd see that the probability the random
variable X takes on the value 1
is equal to p to the 1, (1-p) to the 1-1,
which is just equal to p.
And the probability that the random
variable X takes on the value 0
is equal to p to the 0, (1-p) to the 1-0,
which is just 1-p.
(The probability of success and the
probability of failure that we talked about previously.)
What we have up here is just a way of writing that in one line.
Now it can be shown without too too much difficulty
That the mean of a Bernoulli random variable is equal to p,
and the variance of a Bernoulli random variable is equal to p(1-p).
Approximately one in 200 American adults are lawyers.
Suppose we randomly sample one American adult.
We might ask ourselves: what is the
distribution of the number of lawyers?
We've got a single trial, each person is
a lawyer or they are not,
and the conditions of the Bernoulli
distribution here are satisfied.
So the number of lawyers is going
to have a Bernoulli distribution with parameter p.
And p is going to be simply
the probability that any one individual person is a lawyer,
which is about one in 200.
And if we let the random variable X
represent the number of lawyers in
our sample of size 1,
then the probability that the random
variable X takes on the value little x
is going to be equal to p to the x,
which is one over 200 to the x,
times (1-p), which is (1-1/200), to the (1-x)
and this is for x equalling 0 or 1,
because we're either going to get one lawyer or none.
And if we substituted these values in here,
we'd see that the probability
that we do get
one lawyer is going to be equal to 1/200,
and the probability that we don't get a lawyer
(we get 0 lawyers) is going to simply be
199/200.
Now you might be wondering
what is the point of all of this.
We had that approximately one in two
hundred American adults are lawyers,
so it's certainly not rocket science to say
the probability that we got one lawyer is 1/200.
But saying that something has a
Bernoulli distribution
is a nice concise description
of the situation at hand.
Some other common discrete probability distributions
are built on the assumption of independent Bernoulli trials.
For example here we have the binomial distribution,
the geometric distribution and the
negative binomial distribution.
The binomial distribution is the
distribution of the number of successes
in n independent Bernoulli trials.
The geometric distribution is the
distribution of the number of trials to
get the first success in independent Bernoulli trials,
and the negative binomial distribution is
the distribution of the number of trials
to get the rth success in independent Bernoulli trials.
And so we have a number of these other
distributions that are built on this
assumption of independent Bernoulli trials.
And we'll talk about some of those
distributions a little bit later on.