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amethyst playing a lottery game where he must pick two numbers from zero to nine and then one letter out of the 26 letter in lava bet he may choose the same number both times if his ticket matches the two numbers and 1 letter drawn in order he wins the grand prize and receives ten thousand four hundred and five dollars if just his letter matches but one of both one or both of his numbers do not match he wins the small prize of $100 under any other outcome he loses and receives nothing the game costs him five dollars to play so under any other outcome he loses and receives nothing he has chosen the ticket zero for R so R assuming he bought he's paying the five dollars to play and he picks the ticket zero for R so let's say we define a random variable X and let's say that this random variable is the the net profit the net profit from playing from playing this this lottery game what is the expected from I guess we could even say the expected from the net the the net profit from playing 0 for R so M it's particular ticket right over here so let's just say X is the random variable it's 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 value of that what is the expected net profit from playing 0 for R and I encourage you to pause the video and think through it on your own so let's think about what expected value is it's the probability of each of those outcomes times the net profit from from those outcomes so there's the probability of the grand prize right that is let me do that in that red color so there is the probability of getting the grand prize and now what would times his net payoff from the grand prize what would that be well he gets ten thousand four hundred and five dollars but that's not his net payoff his or his net profit I should say his net profit is what he gets minus what he paid to play so he paid $5.00 to play so this would be so that's that so plus plus the probability of getting the small price probability of the small price times the payoff of the small price which is going to be $100 or times the net profit I guess if you get the small price so you get a payoff of 100 minus U at this pay - you have to pay $5 to play and then finally you have the probability the probability of neither so you're essentially not winning and there in that situation what is the net what is the net profit well in that situation your net profit is negative five you paid five dollars and you got nothing in return so to figure out the expected value just have to figure out these probabilities so what's the probability of the grand prize do that over here probability of grand prize well the probability that he gets the first letter right is one and ten there's ten digits there probability to get the second letter right is one and ten these are all independent and probably 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 so the probability of the grand prize is one in what is this 2600 one in 2600 so this is one in 2600 now what's the probability of getting the small prize probability of the small small prize well let's see he has a one in 26 chance the small prize is getting the letter right getting the letter right but not getting both of the numbers right so he has a one in 26 chance of getting the letter right but this isn't we're not done here just with the one in 26 because he has a one in 20s 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 so we need to do is we need to subtract out the situation the probability of getting the two numbers the getting the and the two numbers right and we already know what that is it's one in 2600 so it's one in 26 minus one and 2600 the reason why I have to subtract out at this 2600 is is just one here's a one in 26 chance of getting this letter right so that includes that includes the scenario where he gets everything 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 and you're in the grand prize case so you essentially have to subtract out the probability that you won the grand prize that 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 the probability of neither well it's it's just kind of you know that's everything else so it would be one minus these probabilities right over here so it would be one minus the probability of the small prize probability of the small minus the probability of the grand these are the possible outcomes so they have to add up to 1 or 100% so this is one minus probability small minus minus probability of large minus probability of large or I said not grand prize grand grand prize so let's fill this in so the probability is a small one this right over using that red too much this right over here is 1 and 26 minus 1 and 26 101 and 2600 and then this right over here is 1 minus the small which is 1 in 26 minus 1 and 2600 minus 1 and 2600 minus 1 in 2600 and this simplifies to let's see this is 1 minus 1 over 26 plus 1 in 26100 plus or minus 1 and 2600 these cancel and you're left with 1 minus 1 and 1 and 26 now why does that make why does make sense well the way you lose the way you get nothing is if you get the letter wrong so if you get you have a 126 chance of getting the letter right and then you're going to be in one of these two categories or you have a 1 minus 126 which is equal to 25 out of 26 you have a 25 and 25 26 chance of getting the letter wrong in which case you get nothing in which case you completely lose so let's just kick our calculator out and calculate this and we'll round to the to the nearest well round to the nearest penny here so let's see it is going to be 120 600 so 1 divided by 26 hundred times let's see 10 thousand 4 5 minus 5 is going to be ten thousand 4 hundred times ten thousand four hundred 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 26 hundred times your net profit for the small prize is 100 minus five which is 95 and then finally plus plus 25 26 so 25 divided by 26 extra all to put parentheses around here just to make it consistent so 25 divided by 26 times that net pay off when you get nothing what you have to pay out five dollars and you've got nothing in return times negative five actually I don't know if it's going to recognize that as time so I'll just write x 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 and $2 and 81 cents if we round up to the nearest penny so this is all going to be equal to two dollars and 81 cents and so this is actually a very unusual lottery game where you a positive expected net profit as a player usually the purpose of operating the lottery the state who are the casino wherever 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 most which is not the case with most lottery games and if by playing you actually expect a $2 and 81 cent net profit