Compare the average value with the expected value.
A
a
13/30
B
b
Average Value for a Possesion: 0.8 Expected Value: 0.66
C
c
See solution.
Practice makes perfect
a
We are given that Cynthia used her statistics from last season to design a simulation using a random number generator. Using this simulation, she wants to predict her score each time she got possession of the ball. Let's take a look at the given frequency table.
Integer Values
Points Scored
Frequency
1-14
0
31
15
1
0
16-28
2
17
29-30
3
2
To state what was the assumed theoretical probability of scoring 2 points in a possession, we need to calculate the number of integer values Cynthia assigned to each score.
Points Scored
Integer Values
Number of Integers
0
1-14
14
1
15
1
2
16-28
13
3
29-30
2
Sum
30
The theoretical probability of scoring 2 points will be the number of integers assigned to this score divided by the the number of all assigned integers.
P(2)=13/30
Cynthia assumed that the theoretical probability of her scoring 2 points is 1330.
b
Now, we are asked to calculate Cynthia's average value for a possession and her expected value. Notice that to calculate the average value we use experimental probabilities, while the expected value requires using theoretical probabilities. Let's start with the average value.
Integer Values
Points Scored
Frequency
1-14
0
31
15
1
0
16-28
2
17
29-30
3
2
Sum
50
The average value AV will be the sum of points multiplied by the corresponding experimental probability, which is the frequency divided by the sum of trials.
AV= 0*31/50+ 1*0/50 + 2*17/50+ 3*2/50Let's simplify the above expression.
The average value for a possession is 0.8. Next, let's calculate the expected value. To do so we will calculate the sum of scores multiplied by their corresponding theoretical probabilities, which will be the number of integers assigned to the score divided by the number of assigned integers in total.
Points Scored
Integer Values
Number of Integers
0
1-14
14
1
15
1
2
16-28
13
3
29-30
2
Sum
30
Now we can write the formula for the expected value EV.
EV= 0*14/30+ 1*1/30+ 2*13/30+ 3*2/30
Let's simplify the above expression.
Let's compare the average and the expected values.
Average Value & < Expected Value
0.8 & < 1.1
The average value is less than the expected value. However, if the number of trials in Cynthia's simulation was more than 30 we could expect that the average value from a simulation would be closer to the expected value.
