If the expected values of two events are roughly the same, you can use the standard deviation to compare them. Higher standard deviation means higher risk, a bigger chance of winning a large amount of money, and a bigger chance of not winning anything.
A
a
$0.29
B
b
4.535
C
c
See solution.
Practice makes perfect
a
To calculate the expected value, we will first calculate the probabilities of winning the prizes. We are told that only 1 ticket wins the first prize, 2 tickets win the second prize, and 5 tickets win the third prize. Let's write that down in a table.
Prize
Tickets With the Prize
First
1
Second
2
Third
5
In total, there are 8 tickets that win a prize. Every other ticket does not win anything. We were told that in total there were 500 tickets, so there are 500-8=492 tickets with no prize. Let's add that information to the table.
Prize
Tickets With the Prize
First
1
Second
2
Third
5
No prize
492
To calculate the theoretical probabilities of winning the prizes, we can divide the numbers of tickets of each prize by the total number of tickets. We will think of the tickets without a prize as tickets with a prize of $0.
Prize
Tickets With the Prize
Total Number of Tickets
Probability
$100
1
500
1/500=0.002
$10
2
500
2/500=0.004
$5
5
500
5/500=0.01
$0
492
500
492/500=0.984
Now, we can calculate the expected value which is what you get if you add up the products of the prizes and their probabilities.
Expected Value= ∑ Prize * P(Prize)
Let's calculate the products of the prizes and their probabilities.
Prize
P(Prize)
Prize * P(Prize)
$100
0.002
$100*0.002=$0.20
$10
0.004
$10* 0.004=$ 0.04
$5
0.01
$5* 0.01=$ 0.05
$0
0.984
$0* 0.984=$ 0
Finally, to calculate the expected value we will add up all the values from the last column.
Expected Value= $0.20+$0.04+ $0.05+$0= $0.29
We found that the expected value of a single ticket, not considering the ticket's cost, is $0.29.
b
Let's recall the formula for the standard deviation σ of a discrete random variable X. In our case, X represents the prize of the raffle.
σ = sqrt(∑ [X-E(X)]^2* P(X))
To calculate the standard deviation, we will first calculate the expression inside the radical, which is called the variance of the random variable X.
σ^2 = ∑ [X-E(X)]^2* P(X)In Part B we found that the expected value E(X) is about 0.29, which we can substitute into the equation.
σ^2 = ∑ [X- 0.29]^2* P(X)
The variance σ^2 is a sum of elements. Let's calculate each element of the sum one at a time. We can find the possible prizes X and their corresponding probabilities P(X) using the table we made in Part A.
Prize, X
P(X)
[X-0.29]^2
[X-0.29]^2 * P(X)
$100
0.002
[100-0.29]^2= 9 942.0841
9 942.08 * 0.002≈ 19.884
$10
0.004
[10-0.29]^2=94.2841
94.284* 0.004≈ 0.377
$5
0.01
[5-0.01]^2= 22.1841
22.1841* 0.01≈ 0.222
$0
0.984
[0-0.29]^2=0.0841
0.0841* 0.0827544≈ 0.083
Now, to find the variance σ^2 we need to sum all the elements in the last column together. Let's do it!
σ^2≈ 19.884+0.377+0.222+0.083
= 20.566
Finally, we can calculate the standard deviation by calculating the square root of the variance σ^2.
We found that the standard deviation of the probability distribution is about 4.535.
c
We are told that the Glee's Club raffle has a standard deviation of 2.2. In Part B we found that the standard deviation of the French Club Raffle is bigger, about 4.535. Therefore, French Club's raffle is riskier, higher chance of both winning a lot of cash and also not winning anything at all. Both clubs have the same expected value, $0.29.
Raffle
Standard Deviation
Expected Value
Glee Club
2.2
$0.29
French Club
4.535
$0.29
The cost of a ticket is $1, which is higher than the expected value. This means that on average we lose money in both raffles. We might as well have some fun and participate in the riskier French Club's raffle to have a bigger chance of winning a big prize. The person that solely wants to win money should avoid both raffles.
