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Okay. In this example, we're going to look at a lottery where we're picking five numbers
from 1 to 35 with no repetition. And it costs a dollar for a ticket. And if you get all
five numbers, you win the jackpot. And with a lot of lotteries, the jackpots vary, so
we're going to estimate it at $50,000. If we pick four, we win $500. And if three of
ours that we picked match, we win $5. Otherwise, if we pick two, one, or zero, we win nothing--
we lost our dollar that we initially paid.
And so for each of these payouts, we still lose that dollar, because we have to pay for
that ticket up front, and you only get to redeem that ticket for the winnings if one
of these three events are matched up.
So what is the expected winnings of a single ticket? Remember when we talk about expected
values, the expected value is simply the sum of the product of outcomes with their probabilities. And so that means
that I need to figure out my-- winning is my outcome, the winning value-- I need to
figure out the probabilities for each of those winnings.
So since it's costing me a dollar, I have to factor that in at some point. So either
we can do it at the front, or we can do it at the end. I typically like to approach these
problems by factoring the cost right up in the front. Otherwise, I have a chance of forgetting
about it. So I'm going to be listing out what my net gains are.
So those are the outcomes of buying a ticket. I can win, we estimated $50,000, but, you
know, we've still got to consider the cost of the ticket. So I'm just going to take that
dollar off anyways; although that $50,000 was an approximation. So if we match all five,
you know, our theoretical gain is $49,999. And so I'm going to have to figure out what
that probability is. So I'll get to that in a moment.
The other net gains... so we look at I win $500. It still cost me a dollar for that ticket,
so it's actually winning $499. I win $5. Well, it still cost me a dollar, so I only won $4.
And then if I lost, it's not that nothing happened. I lost my dollar. So my net gain
there is a -1.
Looking at the probabilities, this is where it's going to require a little more work on
our part. The probability of winning is that I have to choose all 5 winning numbers out
of the 35 possible. So the only way for me to win is that all five that I choose match the five selected. So I'm going to have
to calculate this probability, and then we'll do it for each one of these.
This one's not too bad because it's a combination problem. Out of the five winning, I'm choosing
five of them. So this is 5 choose 5. How many different ways could we choose 5 out of the
35 numbers? So remember we're doing the NCR here? So I've got 35 numbers to choose from,
and I want 5 of them. This ratio gives me my probability of winning that jackpot.
So 5 choose 5, there's actually only one way that happens, and you should verify that with
the formula. And if you're using, not a calculator that can do this for you, but if you're doing
it by hand, this is 35 factorial over 5 factorial times 35 minus 5 factorial. So we have that
the probability of winning $49,999 is 1 over this number.
So if you simplify this, we've got 35 times 34 times 33 times 32 times 31 times 30 factorial
simplified down to the largest factorial in the denominator; 5 times 4 times 3 times 2
times that 30 factorial. We know those 30 factorials are going to keep canceling each
other out. And now I just have to reduce this down.
So we can divide 5 into 35; 4 can go into 32, here, eight times; 3 can go into 33 eleven
times; and 2 can go into 34 seventeen times. So pulling out a calculator, I have the probability
that we win that $49,999 is 1 over this product, 7 times 17 times 11 times 8 times 31, which
is, when you put it into a calculator and multiply that out, we get 324,632.
So I've got one taken care of. I've got to find the probability of these other three.
And this is where it's going to get a little bit more challenging to get those probabilities.
So we're going to look at the $499 winning. So the outcome is that I win $499. What this
means is I match four of the five that are chosen. And so I want to look at how many
ways that's possible out of the 35-- matching 5 out of the 35-- which I already know the
denominator for. So that probability that I win $499 is going to be the number of ways
that this can happen. I'm going to match four of them, but my fifth one doesn't match. Out
of the possible ways that I can choose 5 out of 35, and we already saw that to be 324,632,
so that part's done. What we've got to do is figure out the numerator.
So thinking about it, is, well, I match the four, and then that last one-- there's still
a number of ways that I can choose a ball that doesn't match one of those five. So 5
have been chosen means that 30 of them have not. So 30... I need to choose one of them
not to match. And so I'm using the fundamental accounting principal here, is that I've got
one way for the one not to match out of the thirty and then I have to figure out how many
ways I can match five of those.
And, again, that's a combination problem. So I have 5 choose 4 of them. And so we need
to simplify this. The nice thing about the 30 choose 1 is that you're going to see that
the answer's just going to be 30; 5 choose 4 I have 5 factorial over 4 factorial times
5 minus 4 factorial by the definition. That's 1 factorial, which is 1, and here I have 5
times 4 factorial. So the 4 factorials are going to cancel, so I have 5 times 30 actually
on top. And again, the denominator is still 324,632. So my probability of winning is 5
times 30, so 150, over 324,632.
Now I need to find the probability that we win a net gain of $4. So that means that I
match three of these. So, again, we're going to have to use the fundamental accounting
principal just like we did for matching four. So in this situation, I have-- I win a total
of $4, and that means I match three of the five. So that probability of winning, and
again, that's for a net gain, is the same accounting principal, here. First I have to
figure out, well, how many ways can I match three of the five? So that's 5 choose 3. And
then two of them must not have matched those five, so I have to figure out how many ways
can I choose two from the thirty.
And the denominator's still 5 choose-- choosing 5 out of 35, and we saw, again, that was 324,632.
So I need to simplify these two combinations. So I went ahead and I just kind of wrote the
definition for both of those and realized that we just need to reduce this down to a
lower form.
And so that first term, here, I have 5 times 4 times 3 factorial all over 3 factorial x
that's 2 factorial, which is 2 times 1. The second term, I have 30 factorial, and there's
a 28 on the bottom. So I'm going to take this as 30 times 29 times 28 factorial all over
2 factorial, which is 2 times 1. And, not to forget it, we still have the same denominator,
324,632.
I realized I wrote a 6 there.
So simplifying, I'm just going to go through this quickly. The 3 factorials cancel. The
2 cancels; that gives me a 2 here; the 28 factorials cancel; 2 goes into 30 fifteen
times. That's going to leave me in my numerator 5 times 2 times15 times 29. And so when you
take that product, you get 4,350. So I have my probability of winning a net gain of the
$49,999, $499, $4.
So now we're left with the probability that we just lose the dollar by playing the lottery.
Now, that one can happen in a few different ways; either I only match two, I only match
one, or I only match zero. And that seems, just based on the other work, to be a little
bit cumbersome. And so one of the things that we can remember is that when I add up all
the probabilities, I have to get 1 as an answer.
So the probability that I win, or I have that net gain of a negative one dollar, is 1 minus
the probability that we had the $49,999. So I'm adding these all up together, the $499
and the $4. And that seems it's going to be much easier to calculate. So I'm going to
take the 1 minus each of those probabilities. So I had 1 over 624,632. The probability of
$499 was 150 over that number. And the probability that we had $4 we just found was 4,350 over
324632. So I'll be right back after I simplify it.
This gives me a result of 320,131 over 324,632. And that's just 1 minus the sum of these three
fractions. So you can verify that. So on to my result, my expected value. So my expected
net gain, here, is going to be each outcome weighted by their probably and then summing
up the total. So to save time, I just kind of wrote out that result.
And let me give myself a little more room here. The expected value was, again, each
outcome-- so winning the 449,999 jackpot times its probability, the $499 times its probability,
the $4 times its probability, and the probability of, just, that negative $1 outcome-- so I
had the probably times that negative1 outcome of losing.
And so if I simplify this on my calculator, I get that the expected net gain of playing
the lottery-- so again, I'm paying the dollar to play the lottery-- is going to be the sum
of those products. Now if you want it in a fraction form, it's -177882 over 324632. And
that rounds to roughly-- we'll round this to two places here, because we're dealing
with money-- is that I'm losing, on average, 55 cents per lottery ticket that I purchased.