a To find the expected value, you sum all possible earnings multiplied by their respective probabilities.
B
b Subtract the amount you have to pay from the expected value you found in Part A.
A
a $8.75
B
b No.
Practice makes perfect
a The expected value from one spin depends on the probability of spinning each sector and their respective values. We are given two sector angles: 90^(∘) and 135^(∘). To find the probability of spinning each sector, we need to find the last sector angle. Note that the angles should add up to 360^(∘).
90 ^(∘) + 135 ^(∘) + x = 360 ^(∘)
Let's solve the above equation for x.
Now we know all of the sector angles. The probability of spinning a given sector is the ratio it occupies of the total spinner. We can find these ratios by dividing the sector angles by 360^(∘).
By calculating the product of each sector's probability and respective earnings, we get the expected value from that sector. If we add all of the sector's products, we get the expected earnings from one spin.
Event
P
Earnings
P * Earnings
Expected Earnings ($)
P($20)
3/8
20
3/8( 20)
15/2
P($5)
1/4
5
1/4( 5)
5/4
P($0)
3/8
0
3/8( 0)
0
By adding all of the expected earnings per sector, we can get the expected earnings per spin.
b In Part A we found that the game's expectation value without payment is $8.75. If we have to pay $10 in order to play the game, the game's expectation value is lowered by $10.
$8.75 - $10 = - $ 1.25
This time we expect to lose $1.25 by playing the game. The game is not fair.
