The PTO is selling raffle tickets
to raise money for classroom supplies.
A raffle ticket costs $5.
There is one winning ticket out of the 240 tickets sold.
The winner gets a prize worth $98.
The first question is what is the expected value
to you of one raffle ticket?
Let's answer this question on the next slide.
To begin, the expected value is the average gain or loss
of an event if the event is repeated many times.
We compute the expected value by multiplying each outcome
by the probability of that outcome
and then we add the products.
Let's begin by determining the probability of winning.
Because there is one winning ticket out of 240 tickets sold,
the probability of winning is one divided by 240,
and therefore the probability of losing
is one minus this value,
which is 239 divided by 240.
There are 239 losing tickets out of 240 total tickets.
And now let's determine the gain or loss of winning.
If you win, you win a prize worth $98,
but you still have to buy the ticket for $5.
And therefore if you win, the gain is $98
minus the cost of the ticket, which is $5,
and therefore the gain is really $93.
And then the loss of losing would be the loss
of the price of the ticket, which is $5.
To indicate the loss, we use negative $5.
Now we have the information we need
in order to determine the expected value.
Again, the expected value is equal
to the sum of the products of the outcomes
and the corresponding probabilities.
Let's consider the first outcome of winning.
Again, if you win, you gain $93.
So we have $93.
And we multiply by the probability of winning,
which is one divided by 240.
And then plus the second outcome is that you lose.
If you lose, you lose $5,
so the outcome is negative $5
times the probability of losing,
which is 239 divided by 240.
And now we determine the sum of the products
using the calculator and round to the nearest cent.
So we have 93 times one divided by 240
plus negative five
times 239 divided by 240.
And Enter.
To the nearest cent, we have negative $4.59.
So going back to the first slide,
the expected value to you of one raffle ticket
is negative $4.59,
which means you can expect to lose $4.59
for buying one raffle ticket.
Next, we're asked to calculate the expected value to you
if you purchase 10 raffle tickets.
To calculate this, we multiply the expected value
of negative $4.59 by 10.
So negative $4.59 times 10
is equal to negative $45.90.
So if you buy 10 tickets, you can expect to lose $45.90.
The next question is what is the expected value
to the PTO of one raffle ticket?
Well, if the expected value to you is negative $4.59,
indicating you lose this much per ticket,
then the PTO is going to gain this amount per ticket
or for one raffle ticket.
So the expected value to the PTO
is positive $4.59 for one raffle ticket
because they can expect to gain $4.59
by the sale of one raffle ticket or per raffle ticket.
And the last question is
if the PTO sells 240 raffle tickets,
how much money can they expect to raise
for the classroom supplies?
Well, that's going to be the expected value
of one raffle ticket times 240.
So $4.59 times 240,
which is $1,101.60.
Again, the positive values represent gains,
the negative values represent losses.
I hope you found this helpful.