We need to start our study of probability with a little terminology.
How do we talk about probability? We learn early in the course in the
second week about guarding terms, where people say things like well, it
might be the case that a meteor is going to hit your house so you need meteor
insurance. But it might just possibility and that's
different from probability. If you ask how probable it is that a
meteor is going to to hit your house, the answer is going to be pretty low.
But how do you say how low it is? One way to talk about probability is to
say, the number of times that it might happen out of the number of times that
occur. So, you might say, three times out of
ten, this horse will win the race three times out of ten.
In ten races, it will win three times. You can also do that in terms of three in
ten or simply there's a 30% chance. Three times out of ten, 30% chance, same
thing. But the way we're going to put it for the
rest of these lectures is to talk about a probability of 0.3.
Probabilities range from zero to one, one means it's absolutely certain that it's
true. If you say the probability is one, that
means it's going to happen. Zero means, there's no chance it's going
to happen, it's certain that it won't happen, that it's not true.
So, every probability has to be between them.
Between its absolutely certain that it won't happen and its absolutely certain
that it will happen, when you are somewhere in the middle, you're between
zero and one, you got different levels of probability.
That's the way we talk about it. And probabilities come in many different
kinds. The first kind of probability, we're
going to call a priori probability because you figure it out prior to
experience, prior to any kind of experimentation.
So, take for example a coin. It's got tails, it's got heads.
You flip it and what do you get? Well, you can't see, but that time, I
actually got tails. What's the probability it will come up
tails? We tend to just assume that it's a fair
coin. Which means that the probability of
coming up tails is equal to the probability of coming up heads.
So, if they're equal, when you add then together, you should get one because it's
absolutely certain that you're going to get either heads or tails,
one or the other. And that means that you assume the
probability is 0.5. But all of that is just assumption.
You're just assuming that the outcomes are equally likely.
Now, let's turn from coins to dice, little bit more complicated, six sides.
[SOUND] You roll one of them and if you get a two,
as I just did, then the odds of getting a two when there are six sides, if you
assume they're all equally probable, is going to be one out of six.
What if you roll two of them, [SOUND] like we did when we were discussing the
gambler's fallacy. Oh my gosh, I got seven again.
Pretty cool. [LAUGH] And what's the likelihood of
that? Well, how many ways are there to get
seven on a pair of dies? You can get one and six,
you can get two and five, you can get three and four, but you can also get four
and three, five and two, six and one. So, there's six ways to get a seven on
two dice and there're 36 possibilities, 6 * 6.
So, the probability of getting a seven when
you roll two die are 6 out of 36 or 1/6. But notice that we're just assuming that
these dice are fair. It gets a little more complicated when
you have to give up that assumption. So, let's start with coins.
No tricks up my sleeve. It's a normal coin, heads, tails.
What are the [SOUND] odds of getting heads?
Well, two possibilities, heads or tails. If we assume they're equally probable,
the odds of getting a heads are one in two.
We can figure out the probability just by assuming that heads and tails are equally
likely. And that's a priori probability.
But now, let's get violent. If you take the coin and you put it in
your pliers like this, protect the desk,
then you get your hammer and you start. [SOUND] And you bend the coin,
okay? Good.
See? Now, it's bent.
Okay. Now, let's check the probability again.
Here's our bent coin. Now, we flip it.
[SOUND] What are the odds that it's gong to come up heads?
[SOUND] Heads. [SOUND] Heads.
[SOUND] Heads, pretty good. [SOUND] Heads. Well, I can't get it to do
it. But sometimes, it will come up tails and we don't know how often.
It looks like most of the time it comes up heads, but sometimes it will come up
tails. How often?
We don't know. The only way to figure that out is to
flip it hundreds and hundreds of times and see what percentages come up.
That's because, once you bend the coin, it's not an a priori probability.
You can't just assume that the coin has equal chances of landing heads or tails.
You have to turn it into a statistical probability and look at the frequency of
the actual flips for that particular bent coin.
Here's one more example of the difference between a priori probability and
statistical probability. Okay. So, we've got your normal dice here
and we've just learned that [SOUND] the odds of rolling a seven,
[SOUND] there we go, are 6 in 36,
okay? [SOUND] Right?
Now, what about these? [SOUND] Now, we have ten-sided dice here.
You can see what they look like, 10-sided dies.
What are the odds of rolling, say, double fours on 10-sided dies?
Well, it's going to be 1 in 100 because there are ten options on each one, so
there are a 100 possible combinations so 1 in 100.
What about these dice, 12-sided dice, I love different kinds of dice, don't you?
So, here are the 12-sided dice, okay?
[SOUND] And if you roll them, let me do that again [SOUND] then, oh, I got four
and five that's pretty close. What are the odds of getting four and
four though on two 12-sided dice? Well, it's 1 in 144 and we can do all
that pretty simply by taking the odds of each dice multiplying them together to
get the total possible combinations and only one of them is going to be four and
four. So, the odds are 1 in 144.
It gets a little trickier when we start doing something like this around the dice
with a little weight in it, to fall on the number.
But, of course, we don't know what's inside it.
[SOUND] So, I'm not sure why we would believe that there all equally probable,
right? So, what are the odds of getting a two on
a 6-sided round dice? Might be 1 in 6 but you can't really tell
because you, unless you trust the dice, that is.
And what about this one? This one is my favorite because my son
made it for me. It was so cute.
He was about five years old on a clay class [SOUND] he decided he would make me
a dice. What are the odds of getting a two on this one?
Well, you can see it's kind of irregular. And so, you don't really know what the
odds are of getting a two, unless you do a little experiment.
You can't do a priori probability if the, the dice is not regular.
You have to roll it [SOUND] a hundred thousand times [SOUND] or however many
times [SOUND] in order to figure out how many times it comes up one way or the
other. [SOUND] And finally, last but not least, one of my favorite games, Pass the
Pigs. I highly recommend it.
It's a really fun game. Now, what about this?
You roll the pigs and sometimes you get a snouter, or a leaner, or a jowler, or a
razorback, or a trotter. They're different results and,
ooh, there we go, a double razorback, that's pretty good.
And you roll them, ooh, look at that, a leaning jowler, that's a lot of points
there. And we've, you have to roll them along a
lot of different times in order to get the probabilities.
And we actually did that. We did 1,817 rolls.
We got one side up 1,174 times. Trotter, 150.
Razorback, 451. Snouter 39.
Leaning Jowler, 13. So, what are the odds of getting a
leaning jowler on one roll of a Pass the Pig?
13 out of 1,817. Now, notice that you could never do that
a priori, just by assuming probabilities, you have to do an actual experiment and
look at the frequencies because this probability is not an a priori
probability, it's a statistical probability.
And the nice thing about statistical probability is that you can apply it to a
lot of things besides coins and dice, right?
What's the probability it's going to rain tomorrow?
Well, how do they figure that out? They do a lot of observations and when it
rains, under what kinds of circumstances. And they describe a probability on the
basis of how many times it's rained in similar circumstances in the past.
And what's the probability this batter is going to get a hit in baseball?
Well, they look at his batting average over the previous part of the season
especially under certain circumstances versus right-handed pitchers versus
left-handed pitchers and you got a probability that he's going to get a hit
this time. The same for cricket.
If you've got a bowler, it's a spin bowler, or a speed bowler.
Well, this batter might be better against one
than the other and you can figure the probability against the different types
of bowlers. So, in sports, you use statistical
probabilities a lot and in life, you use statistical probabilities a lot.
When you try to decide what kind of car to buy and you want to know what's the
likelihood that this particular car will break down in the first year, you go to
Consumer Reports. And you see, how many cars of this sort
break down in the first year. So, we use statistical probabilities for
a lot of different areas. Really a priori probabilities are only
applicable when you can assume that the outcomes are equally likely.
Now, there's a third type of probability, subjective probability.
And subjective probability can sometimes be based on evidence, but it doesn't have
to. The idea instead is, subjective
probability is a matter of degree of belief or degree of rational belief,
okay? So, somebody says, how likely is it that the couple that just got married,
that live next door, how likely is it that their marriage is going to last?
Well, how do you know, right? You haven't observed their marriage on
many occasions in the past. You can't assume their marriage is
typical of other marriages. So, how are you going to do it?
Well, people tend to describe probabilities on the basis of what they
take to be rational beliefs given what they know about the couple and what they
know about the circumstances and the pressures and whatever.
And sometimes, their beliefs about the likelihood of something happening are
rational, sometimes they're not. But it's a subjective probability either
way because it's based on basically, a guess.
Now, what we're going to do for simplicity is to talk mainly about a
priori probabilities. But the rules that we're going to discuss will apply also to
statistical probabilities and to rational subjective probabilities.
But it's just easier to talk about a priori probabilities so that's where
we'll focus for the next couple of lectures.