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# Expected profit from lottery ticket

Sal multiplies outcomes by probabilities to find the expected value of a lottery ticket. Created by Sal Khan.

Video transcript

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.