How would you put this in expected value and fair price of ticket?
1000 tickets for prizes are sold for $2 each. Seven prizes will be awarded – one for $400, one for $200, and five for $50. Steven purchases one of the tickets. (Give solutions as decimals accurate to the nearest hundredth.)
a) Find the expected value
Note that we include -$2 as a "negative prize" in the likelihood that
Steven doesn't win at all and loses his $2.
Prizes | Probability | Expectation
$398 | .001 | $0.398
$198 | .001 | $0.198
$ 48 | .001 | $0.048
$ 48 | .001 | $0.048
$ 48 | .001 | $0.048
$ 48 | .001 | $0.048
$ 48 | .001 | $0.048
$ -2 | .993 | -$1.986
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Total expectation = -$1.15
The expected value is -$1.15 or a LOSS of $1.15
That is, if this raffle were held every week, and Steven
played it every time, he would average losing $1.15 per
game.
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b) Find the fair price of the ticket
The easiest way to do this problem is just to pretend that
they give him his ticket price back in the problem above.
So we then just add $2 to the -$1.15 and get $2 - $1.15 = $0.85.
But to work it out:
Let the fair price be $x
We put -x where we put -2 above and
-.993x for the expectation, and 0 for
the Total expectation:
Prizes | Probability | Expectation
$400-x | .001 | $0.40-.001x
$200-x | .001 | $0.20-.001x
$ 50-x | .001 | $0.05-.001x
$ 50-x | .001 | $0.05-.001x
$ 50-x | .001 | $0.05-.001x
$ 50-x | .001 | $0.05-.001x
$ 50-x | .001 | $0.05-.001x
$ -x | .993 | -$0.993x
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Total expectation = $0.000
Add the right column and make an equation:
0.85 - x = 0
0.85 = x
That is, if this raffle were held every week, and the price were
85 cents, and Steven played it every time, he would average
breaking even. That fair ticket price is 85 cents.
Edwin