SOLUTION: The Big Bills are a charitable organization that conducts seven online contests with different combinations of prizes and entry costs.
Arrange the contests in increasing order o
Question 1180347: The Big Bills are a charitable organization that conducts seven online contests with different combinations of prizes and entry costs.
Arrange the contests in increasing order of what the chance to win should be so as to give a zero expected value to a player of that game.
Could you please help me put these in order?
1. Prize offered:$200 cost per entry:$5
2. Prize offered:$300 cost per entry:$6
3. Prize offered:$468 cost per entry:$9
4. Prize offered:$440 cost per entry:$8
5. Prize offered:$315 cost per entry:$7
6. Prize offered:$168 cost per entry:$4
7. Prize offered:$540 cost per entry:$10 Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!
Let
x = probability of winning
w = prize money amount
c = cost to play the game
Now define the events
A = event you win the game
B = event you lose the game
Note how P(A)+P(B) = 1
We can further say:
P(B) = 1-P(A)
and
P(B) = 1-x
Now let
V(A) = net earnings if event A happens (ie if you win)
V(B) = net earnings if event B happens (ie if you lose)
We can say
V(A) = winnings - cost = w - c
V(B) = -c, because you lose c dollars
To recap:
P(A) = x
P(B) = 1-x
V(A) = w - c
V(B) = -c
The expected value will have us multiply the probability values with the corresponding net values. Then we'll add up the results like so:
Expected value = P(A)*V(A) + P(B)*V(B)
Expected value = x*(w-c) + (1-x)*(-c)
Expected value = wx - cx - c + cx
Expected value = wx - c
Lastly, we set the expected value equal to zero and isolate x
wx - c = 0
wx = c
x = c/w
Therefore, the probability to win (x) is simply the ratio of the cost to play (c) and the amount you win (w).
Now let's make a table with 7 rows to correspond to the 7 questions. We'll have three columns to represent the prize offered, the cost, and the win probability needed to get an expected value of zero
Here's what part of the table looks like
The work shown in the third column can be written off to the side, or done as scratch work on another sheet of paper. All that matters really are the values in blue.
Prize
Cost
Probability to Win
200
5
x = c/w = 5/200 = 0.025
300
6
x = c/w = 6/300 = 0.02
468
9
x = c/w = 9/468 = 0.01923 (approximate)
440
8
x = c/w = 8/440 = 0.01818 (approximate)
315
7
168
4
540
10
I'll let you complete the rest of the table. Once you've filled out the third column entirely, you'll then sort the third column from smallest to largest.