Voiceover:Ahmed is playing a lottery game
where he must pick two numbers from zero to nine
and then one letter out of the 26 letter English alphabet.
He may choose the same number both times.
If his ticket matches the two numbers
and one letter drawn in order,
he wins the grand prize and receives $10,405.
If just his letter matches but one or both
of his numbers do not match,
he wins the small price of $100.
Under any other outcome, he loses and receives nothing.
The game costs him $5 to play.
Under any other outcome he loses and receives nothing.
He has chosen the ticket 04R.
Assuming he's paying the $5 to play
and he picks the ticket 04R.
Let's say we define a random variable X
and let's say that this random variable
is the net profit from playing this lottery game.
What is the expected from ...
I guess we could even say the expected
from the net profit from playing 04R,
so Ahmed's particular ticket right over here.
Let's just say X is the random variable,
is the net profit from playing this ticket.
What I want to think about in this video
is what is the expected value of that?
What is the expected net profit from playing 04R?
I encourage you to pause the video
and think through it on your own.
Let's think about what expected value is.
It's the probability of each of those outcomes
times the net profit from those outcomes.
There's the probability of the grand prize.
I can write that, let me do that in that red color.
There is the probability of getting the grand prize
and what would times his net payoff from the grand prize.
What would that be?
Well he gets $10,405 but that's not his net payoff
or his net profit I should say.
His net profit is what he gets minus what he paid to play.
He paid $5 to play.
That's that, plus the probability
of getting the small price times the pay off
of the small price which is going to be $100
or times the net profit I guess if you get the small price.
You get a payoff of a 100
minus you have to pay $5 to play
and then finally you have the probability of neither.
You're essentially not winning
and in that situation, what is the net profit?
Well in that situation your net profit is negative five.
You paid $5 and you got nothing in return.
To figure out the expected value,
you just have to figure out these probabilities.
What's the probability of the grand prize?
I'll do that over here, probability of grand prize.
Well the probability that he gets the first letter right
is one in 10, there's 10 digits there.
Probability he gets the second letter right
is one in 10, these are all independent
and probability he gets the letter right,
there's 26 equally likely letters
that might be in the actual one
so he has a one in 26 chance of that one as well.
The probability of the grand prize is one in 2600.
This is one in 2600.
Now what's the probability of getting the small price?
Well let's see, he has a one in 26 chance.
The small prize is getting the letter right
but not getting both of the numbers right.
He has a one in 26 chance of getting the letter right
but we're not done here just with the one in 26
because this one in 26,
this includes all the scenarios
where he gets the letter right,
including the scenarios where he wins the grand prize,
where he gets the letter and he gets the two numbers right.
We need to do is we need to subtract out the situation,
the probability of getting the two numbers,
getting the letter and the two numbers right
and we already know what that is,
it's one in 2600.
It's one and 26 minus one and 2600.
The reason why I have to subtract out at this 2600
is he has one in 26 chance of getting this letter right.
That includes the scenario where he gets everything right
but the small prize is only where you get the letter
and one or none of these.
If you get both of these
then you're at the grand prize case.
You essentially have to subtract out the probability
that you won the grand prize,
if you got all three of them
to figure out the probability of the small prize.
Now what's the probability of essentially losing?
The probability of neither.
Well it's just kind of that's everything else.
It would be one minus these probabilities
right over here.
It would be one minus the probability
of the small prize.
The probability of the small
minus the probability of the grand,
these are the possible outcomes
so they have to add up to one or a 100%.
This is one less probability small
minus probability of large
or I'll say grand prize.
Let's fill this in.
The probability of the small one, this right over ...
I'm using that red too much.
This right over here is one in 26
minus one in 2600
and then this right over here is one minus the small
which is one in 26 minus one in 2600
minus one in 2600.
This simplifies to let's see,
this is one minus one over 26
plus one in 2600 plus or minus one in 2600.
These cancel and you're left with one minus one in 26.
Why does this make sense?
The way you get nothing is if you get the letter wrong.
You have a one in 26 chance of getting the letter right
and then you're going to be in one of these two categories
or you have a one minus one 26
which is equal to 25 of 26.
You have a 25 26 chance of getting the letter wrong
in which case you get nothing,
in which case you completely lose.
Let's just get our calculator out and calculate this
and we'll round to the nearest penny here.
Let's see, it is going to be one 2600.
One divided by 2600 times let's see,
10,405 minus five is going to be 10,400,
times 10,400, that's your net profit
when you win the grand prize
and then you're going to have plus one divided by 26
minus one divided by 2600
times your net profit for the small price
is a 100 minus five which is 95,
and then finally plus 25 26.
25 divided by 26, actually I'll put parenthesis around here
just to make it consistent.
25 divided by 26 times that net payoff.
When you got nothing, well you have to pay out $5
and you got nothing in return, times negative five.
Actually I don't know if it's going to recognize that
as times so I'll just write times negative five
and let me delete that and we deserve a drum roll now.
We get a expected net profit of playing as
$2.81 if we round up to the nearest penny.
This is all going to be equal to $2.81.
This is actually a very unusual lottery game
where you have a positive expected net profit as a player.
Usually the purpose on operating the lottery,
the state, or the casino, whoever it is,
they're the ones who have the expected net profit
and then the player has the expected net loss
but this actually would make rational sense to play
which is not the case with most lottery games
and if by playing you actually expect a $2.81 net profit.