>>Dr. Holmes: Let’s do a binomial probability problem. Here’s an example worked out step
by step, following the logic of the process we presented earlier. Here’s our question:
The likelihood that someone who logs onto the World Wide Web would purchase an item
is point two. If the site has ten people accessing it in the next minute, what is the probability
that: a) none of the individuals will purchase an item; b) at least two individuals will
purchase an item. So we have the set of questions we discussed earlier. Number one, what is
the experiment? We’re not drawing cards or flipping coins, but in reality this is
just like flipping coins. We assume people on the web are acting independently of each other.
That is if I choose to buy it, it does not effect the decision of someone else to
buy it. Also like flipping coins, there are only two outcomes: buy, not buy; heads, tails.
Number two, what is the random variable? What are we interested in? In this case it is the number
of people who buy. It is a continuous or discrete random variable, which is that?
Random variable is discrete because it is measured in whole numbers of people who buy.
Number three, what values can the random variable take on? There can be no one who buys or one
person who buys or two people, etc. until we get to ten. We should write this down.
The random variable is X number of people who buy. The possibilities are none, one,
two, three, dot dot dot dot dot, up to ten. Now, let’s state formally what is being
asked. For problem A, none of the people buy, translates into math as the probability that
X is equal to zero. Now we go to the general formula for the binomial. We know the binomial
because the experiment matches all the characteristics of the Binomial Process. These characteristics
are: it is a binary process, meaning only success and failure, it has constant probability
of success, meaning independent events, and it is a discrete random variable. Here’s
the general formula for the binomial probability density function: P of X is equal to n combinatorial
X, P raised to the x power, Q raised to the n minus x power. Where Q is equal to one minus
p. Q is the probability of failure, p is the probability of success. Now we can simply
put the numbers into the equation. Probability that X is equal to zero, no one buys. Oops,
sorry there. I’m sorry again….is equal to n, we have ten. And we’re going to have
zero and this is going to be P raised to the zero power, which of course is one, and Q
raised to the n minus p or the tenth power.
We substitute in four point two here and point eight here.
What this allows us to do is develop our answer.
Here’s where a calculator will help us in the tedious task of getting the actual answer.
I’ll let you sort that out and I’ll just tell you that the answer is point one zero
seven four. Okay?
Now to part B. we see right away that we’re going to have to do a lot of calculations.
When we write out the question, we see that we’re asking for the probability of two
people or three or four or five or six, and on and on for ten people. We’re looking
for the probability of all of those people buying. That means we’re going to end up
having to calculate nine times, ok? Now, we can use this information that the sum of all
the probabilities is equal to one. This is always going to be the case for any kind of
a probability density function that the sum of all the values it can take on is equal
to certainty. Now if we use this information what we can do is we can take zero and one
and subtract that from one. So the probability form would look like this: the probability
that X is equal to or greater than two, that’s all these here. Is equal to one minus the
probability that X is equal to zero, plus the probability that X is equal to one. That’s
what it looks like mathematically.
Now we only have to do the calculations. The calculations of course are going to be a little
more easy to do because we’re only doing two of those. With this formula, we can do
the calculations and we can get our answer. Here is our answer: point six two four two. And
I’ll leave it to the interested student to verify that. All probability processes follow
this probability density function. This probability density function has many applications, as
you will see in the upcoming units.