Lesson 7: Understanding Uncertainty

Lesson 7: Understanding Uncertainty

Lesson Overview

Figure 7.1 illustrates key aspects of the Statistical Paradigm.  We are interested in the characteristics of a population or how it might behave under different real-world conditions.  To investigate, we take an appropriate sample or compare samples under different conditions (Lessons 1 and Lesson 2).  Our samples then produce data that must be described and/or compared using appropriate measures, statistical summaries, and pictures (Lesson 3 to Lesson 6). Next, in order to use those statistics to make inferences about the population, we must consider what types of samples or differences between groups might happen just by chance - that is the topic of probability that we examine in this lesson.


Statistical Paradigm

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The rules of probability can tell us the likelihood of different types of samples that might arise from a particular population.

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We want to infer what parameter values are most consistent with the sample statistic at hand.

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What does our knowledge of the parameter values tell us about the population?

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Describe and Compare

Data is collected from the samples and, with sample data in hand, we attempt to create statistical summaries and pictures that give the salient features of the data collected.

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Statistical Summaries and Pictures

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Figure 7.1 Key Components of the Statistical Paradigm


Life is riddled with decisions that must be made in the face of uncertainty.  What time should your friend pick you up at the airport given the uncertainty in the arrival time of your flight? Should you take an umbrella on your hike given the uncertainty in the weather forecast? Is it better to have the surgery or take the new medication given the uncertainty in the clinical outcome? Should you bet on the favorites or go for the underdogs in your office March Madness basketball pool?  Statistics help us to address the myriad of problems of this type by directly quantifying the uncertainties in life.  Probability is the language of certainty.  This lesson provides insights into this language's basic grammar - the rules that govern the quantification of uncertainty.


After successfully completing this lesson you should be able to:

  • Identify and apply the relative frequency interpretation of probability.
  • Apply the basic rules of probability.
  • Set up a calculation for and interpret an expected value.
  • Interpret how proportions and averages converge when the chance process generating them is repeated independently.
  • Set up and interpret a random simulation to estimate a probability.

7.1 - The Rules of Probability

7.1 - The Rules of Probability

Example 7.1

Astragalus bone

The astragalus (ankle or heel bone) of animals were used in ancient times as a forerunner of modern dice. In fact, Egyptian tomb paintings show that sheep astragali were used in board games as early as 3500 B.C. (see Figure 7.2). When a sheep astragalus is thrown into the air it can land on one of four sides, which were associated with the numbers 1, 3, 4, and 6 (see Table 7.1). Two sides (the 3 and the 4) are wider and each come up about 40% of the time, while the narrower sides (the 1 and the 6) each come up about 10% of the time. Astragali were used for gambling, games, and divination purposes by the ancients.

Table 7.1 Results Associated with Rolling an Astragalus
Number % of time it occurs
1 10%
3 40%
4 40%
6 10%

 Toss an astragalus once. What's the chance you get a "1"?


The probability of rolling a "1" is 10%. This corresponds to the percentage of time you would get a "1" in a very long sequence of rolls. Here, we can imagine repeating the process of rolling the astragalus many times and with many rolls, the margin of error shrinks and we hone in on an expected relative frequency of times that the event we are looking for happens (getting a "1"). This illustrates the relative frequency interpretation of probability. If a chance process can be repeated independently over and over, the probability of a particular event of interest is just the long-run proportion of times that it happens.

Since probabilities can often be viewed as the proportion of times something happens we see our first rule of probability.

Rule 1: The probability that something happens must be a number between 0 and 1.

 Toss an astragalus once. What's the chance you get at least a 3?


Getting "at least a 3" corresponds to getting either a 3, a 4, or a 6. These occur 40% +40%+10% = 90% of the time so the probability is 0.9. "Getting a 3" and "getting a 4" and "getting a 6" are mutually exclusive events because you can only have one of them happen at the same time (i.e. knowing one thing happened means the other could not have occurred). Our calculation of the probability of "at least a 3" illustrates our second rule of probability.
Rule 2: If outcomes cannot happen simultaneously, the probability that at least one of them occurs can be found by adding their individual probabilities.

Notice that there is another way to solve the previous problem. The opposite of "at least 3" is "getting a 1" (i.e. the only other possibility) so you can also figure the answer as 100% - 10% = 90% or 0.90. This rule of the opposites is our third rule of probability.

Rule 3: The chance of something is 1 minus the chance of the opposite thing.

 Suppose you toss an astralgus twice. What's the chance that you get "4s" on both tosses?


The probability that you get a 4 on the first toss is 40%. Now, when we toss the second time, the chance is still 40% that you would get a "4". Thus, getting two 4s in a row would occur 40% of 40% = 16% of the time. The calculation is just 0.40x0.40 = 0.16. Notice the assumption built into this calculation. We assume (quite reasonably in this case) that the chances don't change when we are making our second toss, regardless of how the first toss comes out.
Rule 4: If knowing how one event turns out doesn't affect the probability for a second event, then the events are said to be independent. The chance that two independent events both happen can be found by multiplying the chances.

 In ancient Rome, the lowest score in tossing four astragali (getting all four 1s) was called the dog throw. What is the probability of getting a dog throw?


Since each throw will be independent the probability would be 0.1 × 0.1 × 0.1 × 0.1 = 0.0001 or one chance in 10,000.

Example 7.2 Independent or not?

  1. For the next two single births at Hershey Medical Center: whether the first baby is a boy and whether the second baby is a boy.

    Independent. Whether the first baby is a boy doesn't affect the chances on the next birth (since they are two different mothers
  2. For the next two single births at Hershey Medical Center: whether the first baby is a girl and whether both babies are girls.

    Not Independent. If the first baby is a girl then the chances go up for both babies being girls (in fact, if the first baby is a boy then the chances drop to zero for both being girls).
  3. For the end of the month in February next year: whether there will be snow on the ground at the State College airport on February 27th and whether there will be snow on the ground at the airport on February 28th

    Not independent. Knowing that there will be snow on the ground on the 27th makes the chances go up for there to be snow on February 28th.

Example 7.3

The highest paid employee has randomly selected from the list of Fortune 500 companies. Which of these probabilities is the largest?

  1. The chance this person is a college graduate
  2. The chance this person is a college graduate with a Business degree.
  3. The chance this person is a college graduate with an Engineering degree.

Choice 'a' since all of the other choices include the fact that the event in choice 'a' happens. This is an example of our next rule of probability.
Rule 5: If the ways in which one event can happen are a subset of the ways that a second event happen, then the first event cannot have a higher probability.

While we can often think of how the process leading to data might be repeated, some events arise in situations that are not easily seen as being repeatable. In such situations, the relative frequency interpretation of probability may seem inappropriate. For example, answering a question like "What is the chance that our next President will be a woman?" would seem to require a different interpretation of the meaning of probability. Luckily, it is perfectly reasonable to assign probabilities to events outside of the relative frequency interpretation as long as they satisfy the above rules of probability. Personal probabilities that satisfy these rules give a coherent interpretation even if they might differ from one person's assignment to another's.

7.2 - Expectations and the Law of Large Numbers

7.2 - Expectations and the Law of Large Numbers

Example 7.4

The most recent Life Tables for the U.S. Population indicates that the probability that a 63 year old man survives to his 64th birthday is 0.9861. An insurance company sells a \$100,000 one-year life insurance policy to 63 year old men for \$1600. How much money do they make on average per policy sold?

A 63 year old man will die during that year with a probability of 1 - 0.9861 = 0.0139. Thus, for every 10,000 policies sold, you would expect them to collect \$1600 from all 10,000 men (totaling \$16,000,000) and pay out \$100,000 to 139 of them (totaling \$13,900,000). That is an average profit of (\$16,000,000 - \$13,900,000) / 10,000 = \$210.

This is an illustration of the Expected Value, which is the long run average value of a measurement. Here the measurement is the profit made by the insurance company for a random male 63 year old customer. To compute the expected value, you sum over the possible values times the probability of getting that value.

Example 7.5

Among passenger cars traveling on the Pennsylvania Turnpike on Thanksgiving weekend: 50% have only the driver; 30% have 2 people in the car; 10% have three people and 5% have 4 people, and 5% have 5 people. You stop along the turnpike. What is the expected number of people in the next car you see pass you?

To sum the values times the probabilities we get 1(0.5) + 2(0.3) + 3(0.1) + 4(0.05) + 5(0.05) = 1.85 people is the expected value.

The expected value denotes a long run average when the basic chance process is repeated over and over. The Law of Large Numbers indicates how this behavior works.

The Law of Large Numbers:

Averages or proportions are likely to be more stable when there are more trials while sums or counts are likely to be more variable. This does not happen by compensation for a bad run of luck since independent trials have no memory.

Example 7.6

EZ Pass Sign

Among passenger cars traveling on the Pennsylvania Turnpike, 30% use Fast-Pass.

 Which is more likely?

  1. That between 25% and 35% of the next 100 cars pay with a Fast-pass, or
  2. That between 25% and 35% of the next 800 cars pay with a Fast-pass

b. The Law of Large Numbers says you are more likely to be within 5% of what you expect for the larger number of trials.

 Which is more likely?

  1. That more than 35% of the next 100 cars pay with a Fast-pass, or
  2. That more than 35% of the next 800 cars pay with a Fast-pass

a. The Law of Large Numbers says you are more likely to be more than 5% away from what you expect for the smaller number of trials.

 Suppose that I see 10 cars in a row with a “Fast-pass” going through the toll area.

True or False: the Law of Averages says that we are now more likely to see cars without the Fast-pass in the next group of 10 cars.

False (the cars are independent so the chance process is still the same). The Law of Large Numbers does not work by compensation.

Example 7.7

Heart with an Arrow through it

Couples eating at a restaurant on Valentine’s Day spend an average of $50.

Which is more likely?

  1. The average amount spent by the next 10 couples is between \$40 and \$60, or
  2. The average amount spent by the next 40 couples is between \$40 and \$60

b. The Law of Large Numbers says you are more likely to be close to what you expect (here within \$10) for the larger number of trials

7.3 - Simulating Probabilities

7.3 - Simulating Probabilities

The behavior of probability models can also be investigated using computer simulation. For example, we can simply verify any of the probability calculations in sections 7.2 and 7.3 above by simulation. The method is straightforward:

  • A probability model involves a chance process that delivers a number.
  • Have the computer randomly generate many instances of this process until the pattern of interest emerges.
  • The beauty of the simulation method is that the user has control over how many times the process is repeated. The Law of Large Numbers tells us that with enough repetitions the pattern will emerge.

Example 7.8: Reeses Pieces Simulation

Reeses Pieces candies come in brown, orange, and yellow colors. The Reese Pieces applet at the Rossman/Chance Applet Collection website lets you simulate draws from a hypothetical population of Reeses Pieces and take data on the orange candies in your sample(s). For example, suppose you wanted to draw 101 Reeses Pieces from a jar with 52% of the candies being orange. You would enter 0.52 as "Probability of Orange" and 101 as "Number of candies" then hit the "draw samples" button. The figure below shows an example result from the Reeses Pieces applet.

image from interactive simulation

In this example, the applet chose 50 orange candies out of 101 - slightly less than a majority even though the true proportion in the population was 52%. Of course, that's what will happen with a random sample. Sometimes you will get a little more and sometimes a little less than the true percentage. If you asked the applet to take a thousand samples instead of just one, you would get a good feel for the randomness from sample-to-sample.

If we wanted to know the probability that our sample would get a majority of orange Reese's Pieces, we can just draw several thousands of samples and see what proportion of those samples found a majority of orange-colored candies. That's the essence of using simulation to estimate probabilities. Now try this on your own. Go to the link and enter 0.52 as the probability of orange, 101 as the number of candies, and say 10,000 as the number of samples then draw the samples and look at the histogram of the results. Now that you have the samples drawn, you can ask the applet to count how many times you had a majority of orange candies in your sample (e.g. ask for a number of orange candies as extreme as 51). You should get an answer to a proportion of around 65% or 66% (since the exact probability is about 65.7%).

Example 7.9

President Obama received 52% of the popular vote in Pennsylvania in the 2012 election. If you took a simple random sample of 101 randomly chosen Pennsylvania 2012 voters, use simulation to find the chance that a majority of your sample would have voted for the President in the last election.


You can use the Rossman/Chance Reeses Pieces simulator to answer this question too. In fact, you already did! This is just the same chance process as we examined in example 7.8. You are picking a sample of size n = 101 with a probability of 52% for the event you are looking at and you want to know the probability of getting that event a majority of times in your samples. The answer is the same regardless of whether you are talking about the proportion of candies that are orange or the proportion of voters who voted for the President.

7.4 - Test Yourself!

7.4 - Test Yourself!

Think About It!

Select the answer you think is correct - then click the 'Check' button to see how you did.

Click the right arrow to proceed to the next question.  When you have completed all of the questions you will see how many you got right and the correct answers.

7.5 - Have Fun With It!

7.5 - Have Fun With It!

Have Fun With It!

cartoon about probability, "Hoping to avoid the difficulties of using conditional probability, Thomas Jefferson writes the Declaration of Independence."

J.B. Landers ©

1 in 2

lyrics and music ©2012-2013 by Lawrence Mark Lesser

I heard of chance in math class while glancing at Sue from afar;
Ev'ry outcome equally likely, rolling a die or drawing cards.
And that gave me hope: she'll either love me or she won't,
So chances must be 1 in 2 that I'll win Sue!

Now I got me a ticket for the lottery: either I hit jackpot or I don't.
So easy to win: I'm ready to spend, and I'll try hard not to gloat
From my mansion, financed by this 50-50 chance
To make my dream come true - - me and Sue!

But odds were "odd" 'cause who would guess
Having 1 in 2 chances, twice with no success?!

It's just as well we never jelled: I know divorce hits 1 in 2.
And lottery's no draw for me: my ping pong ball days are through (they were numbered)!
Yeah, I'm movin' on - - what I knew was wrong:
There's more to chance I must pursue, like 1 in 2. Will skies be blue? 1 in 2.

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