To calculate the relative frequency of a prize, divide the number of people that won the prize by the total number of participants.
B
b
Make a bar graph using the table you made in Part A. Each bar should represent a prize and the bar's height should be proportional to the corresponding probability.
C
c
Make a simulation of the game show using a random number generator.
D
d
Make a bar graph using the table you made in Part C. Each bar should represent a prize, and the bar's height should be proportional to the corresponding probability.
E
e
The expected value is what you get if you add up the products of the prizes and their corresponding theoretical probabilities.
To upgrade the given table to a relative-frequency table, we need to create a column containing the relative frequencies. Let's calculate the relative frequency for each prize X by dividing the number of winners of that prize by the total number of players, 3200.
Prize, X
Winners (Frequency)
Relative Frequency (Probability)
$100
1120
1120/3200=0.35
$250
800
800/3200=0.25
$500
480
480/3200=0.15
$1000
320
320/3200=0.1
$2500
256
256/3200=0.08
$5000
128
128/3200=0.04
$7500
64
64/3200=0.02
$10, 000
32
32/3200=0.01
b
To create a theoretical probability distribution graph, we can use the table we made in Part A. Let's look at that table again! We will only focus on the sums and their probabilities.
Prize, X
Probability
$100
0.35
$250
0.25
$500
0.15
$1000
0.10
$2500
0.08
$5000
0.04
$7500
0.02
$10, 000
0.01
Now we can make a bar graph. We will assign a bar to each prize. The bar's height should be proportional to the probability of winning the corresponding prize.
c
To construct a relative-frequency table for 50 trials, we need to make a simulation of winning a prize in a game show. We can do that by generating random numbers from 1 to 100, and the outcome will represent a prize. Let's set up a proportion based on the probabilities we found in Part A.
Prize, X
Probability
Represented by
$100
0.35
35 numbers
$250
0.25
25 numbers
$500
0.15
15 numbers
$1000
0.10
10 numbers
$2500
0.08
8 numbers
$5000
0.04
4 numbers
$7500
0.02
2 numbers
$10, 000
0.01
1 number
Now, we can choose exactly which numbers from the interval 1-100 will represent which prizes. The lowest prize $100 has to be represented by 35 numbers, so let's choose 1-35. The $250 prize has to be represented by 25 numbers, so let's chose the next 25 numbers 36-60, and so on.
Prize, X
Probability
Represented by
$100
0.35
1-35
$250
0.25
36-60
$500
0.15
61-75
$1000
0.10
76-85
$2500
0.08
86-93
$5000
0.04
94-97
$7500
0.02
98 and 99
$10, 000
0.01
100
We need to make fifty trials, so let's use a graphing calculator to generate numbers from 1 to 100 fifty times. Start by pushing the MATH button. Then scroll right to the PRB menu and choose the randInt( option.
Since we want to generate numbers from 1 to 100 fifty times, we type 1, 100, and 50 on a graphing calculator, respectively. Then press ENTER to get fifty outcomes.
Let's say that we generated the number 80. This number is in the interval 76-85 which represents the prize of $1000. Therefore, we can add one to the frequency of this prize and repeat the process for all the other randomly generated numbers. Let's look at the example results of the simulation.
Prize, X
Frequency
$100
20
$250
10
$500
11
$1000
6
$2500
1
$5000
1
$7500
1
$10, 000
0
Note that this is a random process, so your result might be different. Now, let's calculate the relative frequencies by dividing the frequencies by the total number of trials, 50.
Prize, X
Frequency
Relative Frequency
$100
20
20/50=0.4
$250
10
10/50=0.20
$500
11
11/50=0.22
$1000
6
6/50=0.12
$2500
1
1/50=0.02
$5000
1
1/50=0.02
$7500
1
1/50=0.02
$10, 000
0
0/50=0
d
To make a graph of the experimental probability distribution, we will use the table we made in Part C. We will only need to look at the prizes and their relative frequencies.
Prize, X
Relative Frequency
$100
0.40
$250
0.20
$500
0.22
$1000
0.12
$2500
0.02
$5000
0.02
$7500
0.02
$10, 000
0
Now we can make a bar graph. The height of each prize's bar should be proportional to the relative frequency of winning the prize.
e
The expected value of a discrete random variable X is a weighted sum of its values. In our case, values of X will represent the prizes of the game show.
E(X)= ∑ [X* P(X)]In other words, the expected value is what you get if you add up all products of the possible prizes and their corresponding probabilities. Now, let's use the table we made in Part A to find the products.
Prize, X
P(X)
X* P(X)
$100
0.35
$100* 0.35=$ 35
$250
0.25
$250* 0.25=$ 62.50
$500
0.15
$500*0.15=$75
$1000
0.10
$1000*0.10=$100
$2500
0.08
$2500*0.08=$200
$5000
0.04
$5000*0.04=$200
$7500
0.02
$7500* 0.02=$150
$10, 000
0.01
$10, 000*0.01=$100
We can calculate the expected value by summing all values from the last column. Let's do it!
E(X)= $35+$62.50+$75+... =
$922.50
The expected value is $922.50.
f
We will calculate the standard deviation of a random variable X whose values represents the prizes of a game show. Let's recall the formula for the standard deviation σ of a random variable X.
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 a random variable X.
σ^2=∑ [[X-E(X)]^2* P(X)]In Part E we found that the expected value E(X) is equal to $ 922.50. Let's substitute that value into the equation.
σ^2=∑ [[X- 922.50]^2* P(X)]
The variance σ^2 is the sum of the elements. Let's calculate one element of the sum at a time.
Prize, X
P(X)
[X-922.50]^2
[X-922.50]^2* P(X)
$100
0.35
[100-922.50]^2=676,506.25
990,025*0.3≈ 236,777.19
$250
0.25
[250-922.50]^2= 452,256.25
452,256.25* 0.25≈ 113,064.06
$500
0.15
[500-922.50]^2=178,506.25
178,506.25 * 0.15 =26,775.94
$1000
0.10
[1000-922.50]^2=6006.25
6006.25* 0.1≈ 600.63
$2500
0.08
[2500-922.50]^2=2,488,506.25
2,488,506.25*0.08=199,080.5
$5000
0.04
[5000-922.50]^2=16,626,006.25
16,626,006.25*0.04=665,040.25
$7500
0.02
[7500-922.50]^2=43,263,506.25
43,263,506.25*0.02≈ 865,270.13
$10000
0.01
[10000-922.50]^2=82401006.25
82401006.25*0.01≈ 824,010.06
Now, we can find the variance by summing all the elements from the last column. Let's do it!
σ^2 = 236,777.19 +113,064.06 + +26,775.94...=2,930,618.75
Finally, we can calculate the standard deviation by calculating the square root of the variance σ^2.
