For project presentations, Mr. Fermat has divided the students in his class
into five groups, designated A, B, C, D, and E. Mr. Fermat randomly
selects the order in which the groups make their presentations. Develop
a simulation to compare the probabilities of group A presenting their
project first, second, third, fourth, or fifth.
Method 1: Selecting by Hand
1. Label five slips of paper as A, B, C, D, and E.
2. Randomly select the slips one by one. Set up a table to record the
order of the slips and note the position of slip A in
the sequence.
3. Repeat this process for a total of ten trials.
4. Combine your results with those from all of your classmates.
5. Describe the results and calculate an empirical probability for each
of the five possible outcomes.
6. Reflect on the results. Do you think they accurately represent the
situation? Why or why not?
Method 2: Selecting by Computer or Graphing Calculator
1. Use a computer or graphing calculator to generate random numbers between 1 and 5. The generator must be programmed to not repeat a number within a trial. Assign A = 1, B = 2, C = 3, D = 4, and E = 5.
2. Run a series of trials and tabulate the results. If you are skilled in programming, you can set the calculator or software to run a large number of trials and tabulate the results for you. If you run fewer than 100 trials, combine your results with those of your classmates.
3. Calculate an empirical probability for each possible outcome.
4. Reflect on the results. Do you think they accurately represent the situation?
Why or why not?
The methods in the mentioned investigation can be adapted to simulate any
type of probability distribution:
Step 1 Choose a suitable tool to simulate the random selection process. You
could use software, a graphing calculator, or manual methods, such as dice,
slips of paper, and playing cards. (See section 1.4.) Look for simple ways to
model the selection process.
Step 2 Decide how many trials to run. Determine whether you need to
simulate the full situation or if a sample will be sufficient. You may want
several groups to perform the experiment simultaneously and then pool their
results.
Step 3 Design each trial so that it simulates the actual situation. In particular,
note whether you must simulate the selected items being replaced (independent
outcomes) or not replaced (dependent outcomes).
Step 4 Set up a method to record the frequency of each outcome (such as a
table, chart, or software function). Combine your results with those of your
classmates, if necessary.
Step 5 Calculate empirical probabilities for the simulated outcomes. The sum
of the probabilities in the distribution must equal 1.
Step 6 Reflect on the results and decide if they accurately represent the
situation being simulated.
Many probability experiments have numerical outcomes—outcomes that can be counted or measured. A random variable, X, has a single value (denoted x) for each outcome in an experiment. For example, if X is the number rolled with a die, then x has a different value for each of the six possible outcomes. Random variables can be discrete or continuous. Discrete variables have values that are separate from each other, and the number of possible values can be small. Continuous variables have an infinite number of possible values in a continuous interval. This chapter describes distributions involving discrete random variables. These variables often have integer values.
Usually you select the property or attribute that you want to measure as the random variable when calculating probability distributions. The probability of a random variable having a particular value x is represented as P(X = x), or P(x) for short.
Example - Random Variables
Classify each of the following random variables as discrete or continuous.
a) the number of phone calls made by a salesperson
b) the length of time the salesperson spent on the telephone
c) a company’s annual sales
d) the number of widgets sold by the company
e) the distance from Earth to the sun
Solution
a) Discrete: The number of phone calls must be an integer.
b) Continuous: The time spent can be measured to fractions of a second.
c) Discrete: The sales are a whole number of dollars and cents.
d) Discrete: Presumably the company sells only whole widgets.
e) Continuous: Earth’s distance from the sun varies continuously since Earth
moves in an elliptical orbit around the sun.