Question 1180347
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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. 
<table border = "1" cellpadding = "5"><tr><td>Prize</td><td>Cost</td><td>Probability to Win</td></tr><tr><td>200</td><td>5</td><td>x = c/w = 5/200 = <font color="blue">0.025</font></td></tr><tr><td>300</td><td>6</td><td>x = c/w = 6/300 = <font color="blue">0.02</font></td></tr><tr><td>468</td><td>9</td><td>x = c/w = 9/468 = <font color="blue">0.01923</font> (approximate)</td></tr><tr><td>440</td><td>8</td><td>x = c/w = 8/440 = <font color="blue">0.01818</font> (approximate)</td></tr><tr><td>315</td><td>7</td><td></td></tr><tr><td>168</td><td>4</td><td></td></tr><tr><td>540</td><td>10</td><td></td></tr></table>
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. 

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