>>Let’s go back to the Bright Idea Incorporated light bulb reliability-testing example. Recall
that there were four forty-watt bulbs, five sixty-watt bulbs and six seventy-five-watt
bulbs at our disposal. And we’re randomly selecting three bulbs out of those total of
fifteen bulbs. What’s the probability that all of the bulbs that we select have the same
wattage? Okay, could happen. Let’s first remember how many of which we have. So we
have four forty-watt bulbs, we have five sixty-watt bulbs, and six seventy-five-watt bulbs. Okay,
and we want to know for an event A. Let’s call that event A here. All three have the
same wattage. Okay? And we want to know what’s the probability that this event occurs. What’s
the probability that all three those bulbs that we pick at random have the same wattage,
okay? There are some different ways that that could occur. And so we need to think about
how can we count the number of ways for event A to happen? Okay? Because this is going to
be a classical probability problem, the probability of A is going to be the number of ways A can
occur and we would divide that by the number of ways to pick three bulbs in general. Okay?
So we need to find both the numerator and the denominator. So let’s think about the
numerator, how many ways can the event A occur? How many ways can we get, can we pick three
at random and they all have the same wattage? Okay, well they could all be forty. They could
all be forty watts. Okay, that is forty and forty and forty. So we’ve picked three bulbs
and they all are forty. Forty and forty and forty. Or we could pick three different bulbs
and they’re all sixty, sixty and sixty and sixty. Or union we could pick three different
bulbs and they would all be seventy-five and seventy-five and seventy-five. Okay, and we
can calculate the number of ways that that could occur. Okay, and so how do we do that?
Let’s first calculate this. What’s the, how many ways are there for us to pick all
three light bulbs that are forty watts, well we have forty of those and we’re going to
pick three of them. Okay, so we’ve got four we’re going to pick the three of those four
forty watt bulbs. That means that from the sixty-watt bulbs we’re picking none. And
from the seventy-five watt bulbs we’re picking none. Okay, here we’re going to add to this,
how many ways can we pick sixty-watt bulbs? Three sixty-watt bulbs. Well that means of
the forty watt bulbs we picked none. None of those three are going to be forty-watts,
of the five sixty-watts we’re going to pick all three, and from the seventy-five watt
bulbs we’re going to pick none. Okay, finally, how many ways are there for us to pick all
three of those at random and they all three be seventy-five watts? Well that means we’re
picking none that are forty watts, we’re picking none that are sixty-watts and we’re
picking all three that are seventy-five watts, okay? So this is the number of ways for A
to occur. Keep in mind here, what is five choose zero, what’s anything choose zero
is just one. So essentially we’ve got four choose three plus five choose three plus six
choose three. That’s the number of ways that we can have three different bulbs picked
at random and they’re all three the same wattage. So what should the denominator be?
How many ways are their for us to pick three bulbs in general out of our four plus five
plus six, fifteen total number of bulbs, that’s just fifteen choose three. Okay? So if we
do this we should get a value of something like point zero seven four eight, okay? Pretty
straight-forward idea. We’re counting how many ways are there for A to occur divided
by the total number of ways that we can pick three at random and that probability: point
zero seven four eight.