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.