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We need to start our study of probability 
with a little terminology. 

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How do we talk about probability? 
We learn early in the course in the 

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second week about guarding terms, 
where people say things like well, it 

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might be the case that a meteor is going 
to hit your house so you need meteor 

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insurance. 
But it might just possibility and that's 

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different from probability. 
If you ask how probable it is that a 

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meteor is going to to hit your house, the 
answer is going to be pretty low. 

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But how do you say how low it is? 
One way to talk about probability is to 

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say, the number of times that it might 
happen out of the number of times that 

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occur. 
So, you might say, three times out of 

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ten, this horse will win the race three 
times out of ten. 

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In ten races, it will win three times. 
You can also do that in terms of three in 

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ten or simply there's a 30% chance. 
Three times out of ten, 30% chance, same 

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thing. 
But the way we're going to put it for the 

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rest of these lectures is to talk about a 
probability of 0.3. 

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Probabilities range from zero to one, one 
means it's absolutely certain that it's 

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true. 
If you say the probability is one, that 

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means it's going to happen. 
Zero means, there's no chance it's going 

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to happen, it's certain that it won't 
happen, that it's not true. 

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So, every probability has to be between 
them. 

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Between its absolutely certain that it 
won't happen and its absolutely certain 

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that it will happen, when you are 
somewhere in the middle, you're between 

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zero and one, you got different levels of 
probability. 

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That's the way we talk about it. 
And probabilities come in many different 

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kinds. 
The first kind of probability, we're 

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going to call a priori probability 
because you figure it out prior to 

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experience, prior to any kind of 
experimentation. 

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So, take for example a coin. 
It's got tails, it's got heads. 

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You flip it and what do you get? 
Well, you can't see, but that time, I 

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actually got tails. 
What's the probability it will come up 

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tails? 
We tend to just assume that it's a fair 

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coin. 
Which means that the probability of 

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coming up tails is equal to the 
probability of coming up heads. 

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So, if they're equal, when you add then 
together, you should get one because it's 

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absolutely certain that you're going to 
get either heads or tails, 

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one or the other. 
And that means that you assume the 

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probability is 0.5. 
But all of that is just assumption. 

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You're just assuming that the outcomes 
are equally likely. 

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Now, let's turn from coins to dice, 
little bit more complicated, six sides. 

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[SOUND] You roll one of them and if you 
get a two, 

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as I just did, then the odds of getting a 
two when there are six sides, if you 

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assume they're all equally probable, is 
going to be one out of six. 

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What if you roll two of them, [SOUND] 
like we did when we were discussing the 

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gambler's fallacy. 
Oh my gosh, I got seven again. 

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Pretty cool. 
[LAUGH] And what's the likelihood of 

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that? 
Well, how many ways are there to get 

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seven on a pair of dies? 
You can get one and six, 

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you can get two and five, you can get 
three and four, but you can also get four 

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and three, five and two, six and one. 
So, there's six ways to get a seven on 

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two dice and there're 36 possibilities, 6 
* 6. 

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So, 
the probability of getting a seven when 

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you roll two die are 6 out of 36 or 1/6. 
But notice that we're just assuming that 

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these dice are fair. 
It gets a little more complicated when 

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you have to give up that assumption. 
So, let's start with coins. 

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No tricks up my sleeve. 
It's a normal coin, heads, tails. 

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What are the [SOUND] odds of getting 
heads? 

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Well, two possibilities, heads or tails. 
If we assume they're equally probable, 

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the odds of getting a heads are one in 
two. 

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We can figure out the probability just by 
assuming that heads and tails are equally 

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likely. 
And that's a priori probability. 

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But now, let's get violent. 
If you take the coin and you put it in 

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your pliers like this, 
protect the desk, 

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then you get your hammer and you start. 
[SOUND] And you bend the coin, 

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okay? 
Good. 

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See? 
Now, it's bent. 

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Okay. Now, let's check the probability 
again. 

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Here's our bent coin. 
Now, we flip it. 

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[SOUND] What are the odds that it's gong 
to come up heads? 

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[SOUND] Heads. 
[SOUND] Heads. 

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[SOUND] Heads, pretty good. 
[SOUND] Heads. Well, I can't get it to do 

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it. But sometimes, it will come up tails 
and we don't know how often. 

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It looks like most of the time it comes 
up heads, but sometimes it will come up 

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tails. 
How often? 

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We don't know. 
The only way to figure that out is to 

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flip it hundreds and hundreds of times 
and see what percentages come up. 

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That's because, once you bend the coin, 
it's not an a priori probability. 

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You can't just assume that the coin has 
equal chances of landing heads or tails. 

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You have to turn it into a statistical 
probability and look at the frequency of 

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the actual flips for that particular bent 
coin. 

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Here's one more example of the difference 
between a priori probability and 

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statistical probability. 
Okay. So, we've got your normal dice here 

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and we've just learned that [SOUND] the 
odds of rolling a seven, 

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[SOUND] there we go, 
are 6 in 36, 

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okay? 
[SOUND] Right? 

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Now, what about these? 
[SOUND] Now, we have ten-sided dice here. 

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You can see what they look like, 
10-sided dies. 

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What are the odds of rolling, say, double 
fours on 10-sided dies? 

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Well, it's going to be 1 in 100 because 
there are ten options on each one, so 

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there are a 100 possible combinations so 
1 in 100. 

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What about these dice, 12-sided dice, I 
love different kinds of dice, don't you? 

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So, 
here are the 12-sided dice, okay? 

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[SOUND] And if you roll them, let me do 
that again [SOUND] then, oh, I got four 

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and five that's pretty close. 
What are the odds of getting four and 

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four though on two 12-sided dice? 
Well, it's 1 in 144 and we can do all 

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that pretty simply by taking the odds of 
each dice multiplying them together to 

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get the total possible combinations and 
only one of them is going to be four and 

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four. 
So, the odds are 1 in 144. 

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It gets a little trickier when we start 
doing something like this around the dice 

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with a little weight in it, to fall on 
the number. 

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But, of course, we don't know what's 
inside it. 

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[SOUND] So, I'm not sure why we would 
believe that there all equally probable, 

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right? 
So, what are the odds of getting a two on 

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a 6-sided round dice? 
Might be 1 in 6 but you can't really tell 

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because you, unless you trust the dice, 
that is. 

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And what about this one? 
This one is my favorite because my son 

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made it for me. 
It was so cute. 

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He was about five years old on a clay 
class [SOUND] he decided he would make me 

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a dice. What are the odds of getting a 
two on this one? 

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Well, you can see it's kind of irregular. 
And so, you don't really know what the 

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odds are of getting a two, unless you do 
a little experiment. 

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You can't do a priori probability if the, 
the dice is not regular. 

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You have to roll it [SOUND] a hundred 
thousand times [SOUND] or however many 

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times [SOUND] in order to figure out how 
many times it comes up one way or the 

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other. [SOUND] And finally, last but not 
least, one of my favorite games, Pass the 

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Pigs. 
I highly recommend it. 

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It's a really fun game. 
Now, what about this? 

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You roll the pigs and sometimes you get a 
snouter, or a leaner, or a jowler, or a 

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razorback, or a trotter. 
They're different results and, 

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ooh, there we go, a double razorback, 
that's pretty good. 

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And you roll them, ooh, look at that, a 
leaning jowler, that's a lot of points 

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there. 
And we've, you have to roll them along a 

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lot of different times in order to get 
the probabilities. 

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And we actually did that. 
We did 1,817 rolls. 

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We got one side up 1,174 times. 
Trotter, 150. 

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Razorback, 451. 
Snouter 39. 

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Leaning Jowler, 13. 
So, what are the odds of getting a 

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leaning jowler on one roll of a Pass the 
Pig? 

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13 out of 1,817. 
Now, notice that you could never do that 

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a priori, just by assuming probabilities, 
you have to do an actual experiment and 

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look at the frequencies because this 
probability is not an a priori 

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probability, 
it's a statistical probability. 

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And the nice thing about statistical 
probability is that you can apply it to a 

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lot of things besides coins and dice, 
right? 

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What's the probability it's going to rain 
tomorrow? 

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Well, how do they figure that out? 
They do a lot of observations and when it 

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rains, under what kinds of circumstances. 
And they describe a probability on the 

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basis of how many times it's rained in 
similar circumstances in the past. 

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And what's the probability this batter is 
going to get a hit in baseball? 

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Well, they look at his batting average 
over the previous part of the season 

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especially under certain circumstances 
versus right-handed pitchers versus 

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left-handed pitchers and you got a 
probability that he's going to get a hit 

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this time. 
The same for cricket. 

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If you've got a bowler, it's a spin 
bowler, or a speed bowler. 

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Well, 
this batter might be better against one 

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than the other and you can figure the 
probability against the different types 

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of bowlers. 
So, in sports, you use statistical 

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probabilities a lot and in life, you use 
statistical probabilities a lot. 

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When you try to decide what kind of car 
to buy and you want to know what's the 

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likelihood that this particular car will 
break down in the first year, you go to 

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Consumer Reports. 
And you see, how many cars of this sort 

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break down in the first year. 
So, we use statistical probabilities for 

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a lot of different areas. 
Really a priori probabilities are only 

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applicable when you can assume that the 
outcomes are equally likely. 

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Now, there's a third type of probability, 
subjective probability. 

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And subjective probability can sometimes 
be based on evidence, but it doesn't have 

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to. 
The idea instead is, subjective 

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probability is a matter of degree of 
belief or degree of rational belief, 

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okay? So, somebody says, how likely is it 
that the couple that just got married, 

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that live next door, how likely is it 
that their marriage is going to last? 

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Well, how do you know, right? 
You haven't observed their marriage on 

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many occasions in the past. 
You can't assume their marriage is 

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typical of other marriages. 
So, how are you going to do it? 

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Well, people tend to describe 
probabilities on the basis of what they 

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take to be rational beliefs given what 
they know about the couple and what they 

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know about the circumstances and the 
pressures and whatever. 

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And sometimes, their beliefs about the 
likelihood of something happening are 

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rational, sometimes they're not. 
But it's a subjective probability either 

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way because it's based on basically, a 
guess. 

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Now, what we're going to do for 
simplicity is to talk mainly about a 

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priori probabilities. But the rules that 
we're going to discuss will apply also to 

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statistical probabilities and to rational 
subjective probabilities. 

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But it's just easier to talk about a 
priori probabilities so that's where 

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we'll focus for the next couple of 
lectures.