b Define a trial and use a random number generator.
C
c Compare the average value from a simulation and the expected value.
A
a 36
B
b See solution.
C
c See solution.
Practice makes perfect
a We are given a board with balloons and asked to calculate the expected value of each throw.
Let the random variable X represent the point value assigned to a region on the given dartboard. The expected value E(X) of each throw will be the sum of the point values of each type of balloons multiplied by the corresponding theoretical probabilities.
E(X) = 25* P_(red)+ 50* P_(blue)+ 100* P_(yellow)
Now we will evaluate the probabilities. Since we are asked to assume that each dart hits the balloon, the theoretical probability of each type of balloon will be the number of balloons of that type divided by the number of all balloons on the board, 25.
Type of Balloon
Number of Balloons
Probability
Red
16
P_(red)=16/25
Blue
8
P_(blue)=8/25
Yellow
1
P_(yellow)=1/25
Let's substitute the probabilities into the formula.
b Now we will design and conduct a simulation for this situation. We can use a random number generator to perform the simulation. Because we expressed the probabilities as fractions with denominators of 25, we will assign the integers from 1 to 25 to represent the probability data.
Red:& 1,2,3,4,5,6,7,8,9,10 & 11,12,13,14,15,16
Blue:& 17,18,19,20,21,22,23,24
Yellow:& 25
Each trial — one generated number — will represent the result of one throw. Let's choose the number of trials to be 50. We will use the random number generator in our graphing calculator. To do so, push the MATH button. Then scroll left to the PRB menu and choose the fifth option, randInt(.
After choosing this option, enter the minimum and maximum values of the set and the number of trials. Next, push ENTER.
Now, we can make a frequency table to show the example results of our simulation. Remember the numbers that we assigned to represent each region when keeping track of the outcomes.
Outcome
Tally
Frequency
Red
||||| ||||| ||||| ||||| ||||| ||||| ||||
34
Blue
||||| ||||| ||||
14
Yellow
||
2
Total
-
50
Using the results from the table, we can calculate the average value of the outcomes. We will multiply the point value by the corresponding experimental probability. An experimental probability is the frequency divided by the number of trials, 50.
c Finally, we can compare the expected value and the average value of the outcomes.
Expected Value &> Average Value
36 & > 35
The average value of the outcomes of our simulation is less than the expected value. Keep in mind that this conclusion depends on the simulated outcomes.
