Lesson 7: Understanding UncertaintyLesson 7: Understanding Uncertainty
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.
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.
- 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 Probability7.1 - The Rules of Probability
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.
|Number||% of time it occurs|
Toss an astragalus once. What's the chance you get a "1"?
SolutionThe 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.
Toss an astragalus once. What's the chance you get at least a 3?
SolutionGetting "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.
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.
Suppose you toss an astralgus twice. What's the chance that you get "4s" on both tosses?
SolutionThe 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.
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?
SolutionSince 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?
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
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).
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 28thNot 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.
The highest paid employee has randomly selected from the list of Fortune 500 companies. Which of these probabilities is the largest?
- The chance this person is a college graduate
- The chance this person is a college graduate with a Business degree.
- The chance this person is a college graduate with an Engineering degree.
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 Numbers7.2 - Expectations and the Law of Large Numbers
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?
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.
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?
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.
Among passenger cars traveling on the Pennsylvania Turnpike, 30% use Fast-Pass.
Which is more likely?
- That between 25% and 35% of the next 100 cars pay with a Fast-pass, or
- That between 25% and 35% of the next 800 cars pay with a Fast-pass
Which is more likely?
- That more than 35% of the next 100 cars pay with a Fast-pass, or
- That more than 35% of the next 800 cars pay with a Fast-pass
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.
Couples eating at a restaurant on Valentine’s Day spend an average of $50.
Which is more likely?
- The average amount spent by the next 10 couples is between \$40 and \$60, or
- The average amount spent by the next 40 couples is between \$40 and \$60
7.3 - Simulating Probabilities7.3 - 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.
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%).
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.
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!
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.