Hey it’s Professor Dave, I wanna tell you about probability.
Probability is a mathematical concept that bleeds into our every day lives.
We make decisions all the time based on our perceived probability of success and happiness.
While the probabilities that you will excel at your job or be happy in your relationship
are very impractical to calculate, it’s another matter entirely whether these kinds
of questions are fundamentally impossible to answer.
Let’s go over exactly what probabilities are and how to calculate them, so that we
can better understand how we make predictions.
Let’s take something very simple, like the flip of a coin.
When you flip a coin and call it in the air, predicting heads instead of tails, you are
making a prediction about what will happen.
We can assess the probability of that event happening by taking a number that represents
your desired outcome, and dividing it by a number that represents all the possible outcomes.
In this case, there are two possible outcomes, heads and tails.
Only one of these represents your desired outcome.
That means the theoretical probability of success, if the coin is fair, is one out of
two, which we represent as a fraction.
This fraction could also be expressed as a decimal, like 0.5, which could also be expressed
as a percentage, like fifty percent.
All three of these are saying the same thing, that there’s a one in two chance, or a fifty
percent probability, that you will win this coin toss.
What if we were to flip the coin twice, and you predicted that it will be heads both times?
Now there are four possible outcomes.
We could get heads heads, heads tails, tails heads, or tails tails.
The two coin flips are two separate events that do not affect one another.
The probability of getting heads or tails with a coin flip is always fifty-fifty, no
matter how many times you do it.
So for the first flip, it’s one in two, and for the second flip, it’s one in two again.
One half times one half is one fourth, so that’s 0.25, or twenty-five percent probability
of success.
This is the same as saying that heads heads is one out of the four possible results for
the series of events, which would allow us to arrive at one fourth straight away.
If something absolutely won’t happen, the probability is zero.
If something absolutely will happen, the probability is one.
Anything else must be some value between zero and one, so this is the interval that is allowed
for probabilities.
Sometimes the desired outcome isn’t limited to a singular event.
Let’s say you have a bunch of socks in a drawer.
Five are red, three are blue, seven are green, and ten are yellow.
If you reach in and pull one out at random, what is the probability that it will be red?
In this case, the number of outcomes that will produce the event of getting a red sock
is five, because there are five of them.
Then we divide that by the total number of socks, which is twenty-five.
That gives us 0.2, or twenty percent.
What about the probability of pulling out two socks and getting a red pair?
Well as we said, it’s five out of twenty-five for the first one, but what about the second one?
Now there are four red socks, and only twenty-four socks left to choose from, so that’s a four
out of twenty-four probability for the second event.
We multiply these together, and we get 0.033, or a 3.3 percent probability.
Not the best odds, by any means.
Some of the best examples of odds and probability can be found in casino games involving dice
and cards.
To see some examples of these, let’s visit our friend Heather from Vegas Aces.
Thanks, Dave.
Let’s start by rolling some dice.
Here’s a pair of standard dice, which have values on each of the six sides, from one
to six.
If we roll these dice, what is the probability of getting a twelve?
Well this one’s easy, because there is only one possible outcome that will satisfy this
desired event, and that’s double sixes.
It’s a one in six chance that one of these will be a six, and that the same thing will
happen for the other, so one over six times one over six means a one in thirty-six chance
of rolling a twelve.
But what about the probability of getting a sum of eight or more?
This is more complicated, because there are multiple combinations that will produce this
outcome.
We could generate a grid of all the possible dice combinations, with values increasing
to the right and down, and then we can highlight all the combinations that satisfy this desired
outcome.
That we would be these five that equal eight, these four that equal nine, the three that
equal ten, and the two for eleven, and that leaves the one for twelve we mentioned before.
Five, plus four, plus three, plus two, plus one, equals fifteen out of the thirty-six
total possibilities, or about a 41.7 percent chance.
What are some other ways to think about combinations?
How about a combination lock.
If there are three digits on the lock, each with ten possible values, then we have ten
possibilities for the first number, ten for the second number, and ten for the third number,
and ten cubed is one thousand.
So there are one thousand possible combinations for this lock, and in general, it’s the
number of possibilities per item, raised to the power of the number of items.
This gives us the number of permutations, or specific sequences possible when values
can be repeated, like the way we could use the same number for each digit of this combination.
But what if values can’t be repeated?
Say we have ten people running a race, and we want to find out the number of possible
outcomes for the top three finishers.
There are ten possibilities for the winner, but whoever wins can’t also get second,
so there are only nine possibilities for second place, and eight for third.
We might notice that this is starting to look kind of like those factorials we just learned
about in the previous tutorial.
What if we make this ten factorial instead?
The only problem is that ten factorial would go all the way to one, but we want to stop
after eight, so we have to put this over seven factorial.
That way from seven to one, all the numbers cancel.
So when we want to find the number of possible permutations when values can’t be repeated,
we find N factorial over the quantity N minus R factorial, where N is the total number of
possible items, and R is the number of items in the permutation.
In this case, we want the possibilities for the top three runners out of ten, so R is
three and N is ten.
If we look at card games, things get even trickier, since there are fifty-two cards
in a deck.
Let’s say we want to analyze the probability of getting dealt a particular five-card hand
in poker.
The first thing we need to do is figure out how many five-card hands are possible.
For the first card you get, there are fifty-two possibilities.
For the second card, there are fifty-one possibilities, since it could be any card besides the one
you just got.
For the third, fifty, then forty-nine, then forty-eight.
Just like with the runners, we can represent this as fifty-two factorial over forty-seven
factorial.
Again, this is so that from forty-seven until one, all the numbers cancel, leaving us with
just these five, so we can see that we are again using N factorial over the quantity
N minus R factorial.
But there’s one difference in this case.
This expression gives us the number of possible permutations for the five cards.
Permutations involve a specific order for these items, just like the specific order
of the top three runners.
But when we play cards, the order of the cards doesn’t matter, we are just looking at specific
combinations, so the probability of getting a particular hand increases by five factorial,
because there are five factorial ways for the exact same hand to be distributed.
Since these five factorial permutations all qualify as the same combination, it means
we have to place five factorial in this denominator, thus reducing the number of possible hands,
by accounting for the all of the permutations that result in the same combination.
Thus we have a new expression to find the number of combinations possible when values
can’t be repeated, it’s N factorial over R factorial times the quantity N minus R factorial.
If we evaluate this expression for the number of possible five-card combinations, we get
this number, a little over two and a half million.
Now to look at specific poker hands, let’s check back with Heather.
A standard deck has four suits; spades, clubs, hearts, and diamonds.
Each suit consists of ten numerical cards, from ace through ten, and three face cards,
jack, queen, and king.
From this information, we could calculate the probability of getting a specific poker hand.
Let’s start out with the best hand you could get, a royal flush.
That’s a ten, jack, queen, king, and ace, all of the same suit.
There’s only one of these per suit, and there are four suits, so there are only four
possible hands that result in a royal flush.
Four divided by this huge number is an incredibly tiny number, which even as a percentage is
minuscule.
Basically, if you get one of these in your lifetime, you’re extremely lucky.
Now, what about just a straight flush?
This is like a royal flush, in that the cards have to be in order and of the same suit,
but they don’t have to specifically be a ten through ace.
We could have an ace through five, a two through six, all the way up to a nine through king,
but not a ten through ace, as that’s a royal flush.
That means that there are now nine possible straight flushes per suit, times four suits,
which means thirty-six possible straight flushes.
Again, divided by the total number of possible hands, this is still a super tiny probability.
But what about a four of a kind?
Well there are thirteen possible four-card combinations that would give us a four of
a kind.
We could have four twos, four threes, all the way up to four aces.
But this fifth card can be anything except the other four cards, which means that there
are forty-eight possibilities for the fifth card.
So it’s thirteen times forty-eight, or six hundred and twenty-four possible four of a kinds.
Again we divide that by the total number of possible hands, and it’s still quite small,
but things are getting a little more likely.
How about three of a kind?
Again, there are thirteen possible types of cards from an ace to king, so there are thirteen
types of three of a kinds.
But each type can be one of four combinations of suits, because if we have three kings,
we can have all the kings but the king of spades, all but the clubs, all but the hearts,
or all but the diamonds.
So thirteen times four is fifty-two possible three card combinations to get a three of
a kind.
The fourth card can be anything except these three cards or the fourth card of that type,
since that would make it a four of a kind, which is a different hand, so there are forty-eight
possibilities for the fourth card.
The fifth card can be anything but these three cards, the fourth of that type, this fourth
card, or one of the other three cards of this type, as that would give us a full house.
So there are only forty-four possibilities for the fifth card.
That gives us forty-eight times forty-four, but then we have to divide by two as these
two cards could come in either order.
Multiply that by the fifty-two possibilities for the other three cards, and we get fifty-four
thousand nine hundred and twelve possible three of a kinds.
Dividing by the total possible hands, we get a little over two percent, which is starting
to approach the realm of likelihood.
See if you can calculate the probability of getting a full house, a flush, or a pair,
by using this kind of logic.
So we should now understand permutations and combinations, and the ways they allow us to
compute the probability of certain things happening.
The extent to which this can be applied to complex events involving the stock market
and human behavior is up for debate, but at least for the purposes outlined here, let’s
check comprehension.