>>So let’s work out a few examples that involve counting methods and then translate
those into probabilities. First, say we have a box of bolts and a box of nuts. The box
of bolts contains ten thick bolts, six medium bolts and four thin bolts. Could happen. Then
a box of nuts, let’s say that contains, that has seven that fit the thick bolts, five
that fit the medium bolts and three that fit the thin bolts. One bolt and one nut are chosen
at random from each box, what is the probability that the nut fits the bolt? Okay, big time
problem. You’re stumbling around at night, you’ve got a box full of bolts, a box full
of nuts, you pick one at random, it’s dark, what’s the probability that those two things
fit together, okay? Well this is a classical probability and we’ll see that because we’ve
got this event A, let’s call it event A and that is the event that the nut that’s
drawn at random fits the bolt. Okay? And we are able to count the number of ways that
A occurs, and we’re able to count the total number of ways that any connection of nut
and bolt could occur, okay? So that’s a classical probability, the probability of
A occurring is the number of ways for A to occur divided by the total number of ways
to draw one nut and one bolt. Okay, regardless of whether they fit each other or not, okay?
Could happen. So the denominator is easy to calculate, let’s do that one first. Okay,
so let’s figure out the denominator here. Okay, we have how many nuts? Number of nuts
equals fifteen, okay? Number of bolts, number of bolts in the box? That was twenty. Okay,
so how many different pairs can we make? Well we first take one nut, how many ways are there
to do that? Fifteen. And then we take one bolt, how many ways are there to do that?
Twenty. So fifteen times twenty, three hundred different combinations, three hundred different
ways to draw one nut and one bolt at random from these two boxes, okay? Now let’s talk
about the numerator. This is a little bit more difficult. Not too bad though, let’s
think about the number of ways that we can match different pairs and let’s think about
them from the thick, medium, and thin bolt perspective, okay? So let’s think about
the number of matching thick pairs, okay? So we’ve got a thick bolt and the nut that
goes with that. Well we said that there were ten bolts that fit this description and seven
nuts that fit that description, that would be seventy different matching pairs that would
get us a matching pair of thick, for a thick bolt. Okay? Let’s now think about the number
of matching, matching medium pairs. Okay, we said that there were six bolts, six medium
bolts and five nuts that match those medium bolts. So we have thirty different ways to
pick out a medium nut, and a medium bolt and a matching medium nut. Okay. Then finally
we’ve got the number of matching thin bolt pairs. Okay, we said that there were four
thin bolts and an associated three nuts that fit those four thin bolts, okay? That would
give us twelve different matching pairs, twelve different ways to match a thin bolt and a
nut that fits a thin bolt. Okay? So what are the different ways for us to get a matching
pair, a nut and a bolt that fit together? Well we could have these seventy, or these
thirty, or these twelve. Okay, So the or implies plus, so seventy plus thirty plus ten. Seventy
plus thirty, plus ten, or pardon me, twelve. Seventy plus thirty plus twelve which is one
hundred and twelve. So in the numerator we have a hundred and twelve. A hundred and twelve
ways to get a matching nut and bolt and three hundred different ways to get any kind of
nut and bolt whether it regardless to whether it matches. So we’ve got one hundred twelve
in the numerator, three hundred in the denominator and if we work this out that is point three
seven three three. Okay? So again we’ve got a box of bolts, a box of nuts with these
characteristics, what’s the probability that just at random whenever we pick one form
one box and one from another, what’s the probability that those two fit together? Point
three seven three three.