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Math Lecture Notes
Likelihood

In-Class Activity link to here link to index

Fair Spinner Games

plastic spinner with a clear base

We get to play with more toys. Yay!

Each group of students shares one plastic spinner with a clear base.

We can put a spinner onto a circle like the one below to make a game.

The first circle is a very boring game. Half the time a player wins $1. Half the time a player loses $1.

a simple fair spinner game

What we want to focus on is that the game is fair. This means that if someone played it a whole lot—long enough for any rare streaks of good or bad luck to cancel out—then overall they would not gain or lose money.

1. How can we draw lines to make a spinner game fair if winning has twice the gain of losing?

a second fair spinner game

1. Pretend the circle is a pie, and we are cutting the pie into slices.

You could think that the +2 needs to happen half as often as the −1.

Or you could think that the −1 needs to happen twice as often as the +2.

Either way, we should give the −1 two pie slices, but only give the +2 one pie slice.

That is a total of 2 + 1 = 3 pie slices. So we cut the pie into thirds. The result looks like this:

a third fair spinner game

2. How can we draw lines to make a spinner game fair if winning has three times the gain of losing?

a third fair spinner game

2. Pretend the circle is a pie, and we are cutting the pie into slices.

You could think that the +3 needs to happen one-third as often as the −1.

Or you could think that the −1 needs to happen three times as often as the +3.

Either way, we should give the −1 three pie slices, but only give the +3 one pie slice.

That is a total of 3 + 1 = 4 pie slices. So we cut the pie into quarters. The result looks like this:

a third fair spinner game

3. How about this spinner game with three possible outcomes? Can it be made fair?

a fourth fair spinner game

3. Pretend the circle is a pie, and we are cutting the pie into slices.

If we give the +1 and +2 each one slice of pie, the slice for the +2 counts double. We effectively have three slices of pie for winning +1.

This means we need three slices of pie for the −1.

The two winning numbers each get one slice. The −1 needs three slices.

That is a total of 2 + 3 = 5 pie slices. So we cut the pie into fifths. The result looks like this:

a third fair spinner game

Facts link to here link to index

Probability and Odds

Probability of an Event

The probability of a situation happening is the ratio of desirable outcomes to total outcomes. (This ratio is often changed into percent format.)

Problems that involve probability almost always involve a bunch of counting. Usually there are no convenient formulas to help us. We need to make lists or tables to count the outcomes.

A classic example of probability is rolling two dice and adding their values.

table showing the sum of two dice

4. When rolling two dice, what is the probability of the sum being seven?

4. Looking at the green boxes on the chart, we see that six out of thirty-six possibilities have a sum of seven. So the probability is 636, which we should reduce to 16 or change to about 16.7%.

5. When rolling two dice, what is the probability of the sum being ten?

5. Looking at the pink boxes on the chart, we see that three out of thirty-six possibilities have a sum of seven. So the probability is 336, which we should reduce to 112 or change to about 8.3%.

Imagine there is a gumball machine with equal amounts of three colors of gumballs: red, green, and blue. The table below shows all twenty-seven possibilities for getting three gumballs.

table of possibilities for three gumballs of three colors

6. If you get three gumballs, how likely is it to get at least one blue gumball?

6. Nineteen of the twenty-seven possibilities have at least one blue gumball. So the probability is 19/27 or about 70%.

Odds of an Event

The odds of a situation happening is the ratio of desirable outcomes to undesirable outcomes. (This ratio is often reduced, but never changed into percent format.)

7. When rolling two dice, what are the odds of the sum being seven?

7. Looking at the green boxes on the chart, we see that six out of thirty-six possibilities have a sum of seven, and thirty do not. So the odds are 6 to 30, which could be reduced to 1 to 5.

8. When rolling two dice, what are the odds of the sum being ten?

8. Looking at the pink boxes on the chart, we see that three out of thirty-six possibilities have a sum of ten, and thirty-three do not. So the odds are 3 to 33, which could be reduced to 1 to 11.

9. If you get three gumballs, what are the odds of getting at least one blue gumball?

9. The odds of getting at least one blue gumball are 19 to 8, which can be reduced to 2 to 1.

In this class we will always write ratios using the word "to". For example, 1 to 5. Other math books, websites, and real-life contexts might use a colon instead, and write the same ratio 1 : 5.

Probability of a Change—Absolute Change

Absolute and Relative Change

Unfortunately, there is vagueness about how to measure the probability of a change.

Imagine a medicine that can reduce one of your family member's cancer risk from 44 cases among 10,000 people down to 11 cancer cases among 10,000 people. The medicine has some bad side effects. Is the reduction in cancer risk worth suffering these side effects?

10. We could use subtraction to find the absolute change in risk.

Example 10 (Absolute Change)

The old risk is 0.44%. The new risk is 0.11%.

0.44% − 0.11% = 0.33%

We could say that with the medication the risk is reduced by 0.33%. (Only a third of one percent? That does not sound like much.)

Probability of a Change—Relative Change

We could also use a percent change to measure how much less likely is the new risk. Like all percent changes, this is a ratio comparing change to original.

(The funky part of this problem is how the change and original amount are both percentages.)

11. Using a percent change to measure "less likely" is a relative change.

Example 11 (Less Likely Relative Change)

The old risk is 0.44%. The new risk is 0.11%. Subtracting tells us the change is 0.33%. Then we do change ÷ original.

0.33% ÷ 0.44% = 0.75 = 75%

We could say that with the medication the occurrence of cancer is 75% less likely than before. (That sounds impressive!)

We could use a ratio to talk about whether the new risk is as likely as the old risk. Since this is another ratio, it is also called a relative change.

Example 11 (As Likely Relative Change)

The new risk is 0.11%. The old risk is 0.44%.

0.11% ÷ 0.44% = 0.25 = 25%

We could say that with the medication the occurrence of cancer is only 25% as likely than before. (That still sounds impressive!)

The moral of the story is to pay attention (especially when dealing with small numbers) to whether a speaker is using an absolute change or a relative change. The former made the medicine sound like it is probably not worth the risk of its side effects. The latter made the medicine sound amazing.

Formulas link to here link to index

Weighted Average

measuring spoons

The weighted average of a group of situations measures the "average result" of that group.

To find an weighted average, use a table. Each possible outcome is a row. Work across with multiplication: the value for that outcome times its percent probability. Then add those products.

12. When rolling two dice, what is the weighted average?

12. This answer requires making a table, as below. The answer is not surprising. Most people already know that the "average value" when rolling two dice is seven. The expected value table confirms that common knowledge is precise instead of rounded: the expected value is indeed seven exactly, not slightly more or less. (The original Google spreadsheet is here.)

expected value of two dice

13. Every time little Billy is taken to the grocery store he takes three pennies for this gumball machine. He wants a blue gumball and will spend up to three cents trying to get one. What is the average number of pennies he spends?

13. Here is a sample spreadsheet that shows Billy spends an average of about 2.1¢ each trip to the grocery store.

table to find expected value for gumballs

For many students the most commonly used weighted average table is finding their overall grade in a class.

14. A student has earned the grades below. What is the student's overall grade in the class?

table to find expected value for a grade

14. Here is a sample spreadsheet that shows the overall grade is 81.9 in the class.

table to find expected value for grades

The weighted average is sometimes called the expected value. It does make sense to say "the expected value of the sum of two dice is 7". It almost makes sense to say "the expected value of one of Billy's trips to buy a gumball is 2.1¢." But it does not make sense to call the overall class grade an "expected value" because that situation does not involve condensing mutually exclusive outcomes into an average outcome.

Homework link to here link to index

More Practice

table showing the sum of two dice

15. When rolling two dice, what is the probability of the sum being 8 or more?

15. Looking at the chart, we see that fifteen out of thirty-six possibilities have a sum of eight or more. So the probability is 1536, which we should reduce to 512 or change to about 42%.

16. When rolling two dice, what are the odds of the sum being an even number?

16. Looking at the chart, we see that fifteen out of thirty-six possibilities have a sum of eight or more, and twenty-one do not. So the odds are 15 to 21, which we should reduce to 5 to 7.

17. When rolling two dice, what is the probability of the sum being an even number?

17. Looking at the chart, we see that eighteen out of thirty-six possibilities have a sum that is even. So the probability is 1836, which we should reduce to 12 or change to about 50%.

18. When rolling two dice, whatare the odds of the sum being an even number?

18. Looking at the chart, we see that eighteen out of thirty-six possibilities have a sum of eight or more, and eighteen do not. So the odds are 18 to18, which we should reduce to 1 to 1.

19. The medicine trastuzumab, which fights breast cancer in women who already have breast cancer, was popularized because of a certain study. In the control group of 1,700 women, 34 died. In the group treated with trastuzumab, 23 of 1,643 women died. What percentage of the women in the control group died? What percentage of the women in the treated group died?

19. For the control, group, 34 ÷ 1,700 = 0.02 = 2%.

For the treated group, 23 ÷ 1,643 = 0.0139 ≈ 1.4%.

20. Continuing the previous problem, what was the absolute change (subtraction) in risk? What was the relative change (percent change) in risk?

20. The absolute percent change is 2% − 1.4% = 0.6%.

The relative percent change = change ÷ original = 0.6% ÷ 2% = 0.3 = 30%.

21. Continuing the previous problem, if you were in charge of publicity for this drug, what type of claim could you truthfully make about the medicine?

21. You could say, "People that take this drug are 30% less likely to die from breast cancer."

22. Continuing the previous problem, Trastuzumab also has some dangerous side effects. Most notably, 40% of the women who take it develop flu-like symptoms, 7% develop mild heart problems, and 5% suffer a stroke or severe heart failure. If you were trying to discredit trastuzumab (perhaps concerned about the side effects and trying to convince a family member with breast cancer not to take the medicine) what type of claim could you truthfully make about the medicine?

22. You could say, "People that take this drug only have their cancer risk reduced by less than 1%, but have a 5% chance to suffer side effects of stroke or severe heart failure."

23. You need to buy health insurance, and you talk to some insurance agents and get some probabilities for people like you. Two plans have similar coverage. Plan A has a high price and medium deductible. You will surely pay $500 per month for the insurance, and during the year are expected to pay 90% of the $300 deductible. Plan B is the opposite. You will surely pay $350 per month for the insurance, and during the year are expected to pay 60% of the $2,500 deductible. Which plan is more affordable?

23. Plan A is expected to cost ($500 × 12) + (0.9 × $300) = $6,270.

Plan B is expected to cost ($350 × 12) + (0.6 × $2,500) = $5,700, which is a better deal.

24. At a carnival, a booth offers a dice game in which you roll two dice and find the sum. If the sum is even the booth operator will pay you $1. If the sum is odd you will pay him $1. Is this a fair game?

24. Yes, the game is fair. The expected value table has two rows, and the sum of the products is zero. Overall you will gain and lose the same amounts, and end up at zero.

ResultValueProbabilityProduct
even+10.5+1 × 0.5 = +0.5
odd−10.5−1 × 0.5 = −0.5

25. Your little brother thinks that ten is a very big number. He wants to play a dice game about the number ten. He proposes a game where you each start with a pile of candies, and he finds the sum of two dice several times. Whenever the sum is less than ten, he gives you one candy. Whenever the sum is ten or greater, you give him more than one candy—but he is not sure how many is fair. Help your brother finish inventing his game by using an expected value table to find how many candies must you give him when he "wins" so that the game has an expected value of zero.

25. We need to make an expected value table. We will do it from your little brother's point of view, so giving you a candy is negative but receiving candy is positive.

ResultValueProbabilityProduct
less than ten−130363036
ten or more+y636(6 × y)36

We want the game to be fair, which means the sum of the products should be zero. Therefore the numerators of the two products should match (but one is negative and one is positive). So 30 = 6 × y. Each time your brother wins you should give him 5 candies.

26. You are going to be the instructor of a Biology class and need to write the syllabus. You want attendance to be worth 10%, the final exam to be worth 30%, and the rest divided between nine lab reports and two midterms with each midterm worth as much as three lab reports. How much is each lab report worth? How much is each midterm worth?

26. Be careful, this is not an expected value problem! All we are trying to do is have the graded values add up to 100%.

The nine lab reports and two midterms need to sum to 100% − 10% − 30% = 60%.

Since each midterm should be worth three lab reports, that means we have a total of 9 + 3 + 3 = 15 lab report equivalent things to together make up that 60%.

So each is worth 60% ÷ 15 = 4%.

In other words, lab reports are worth 4%. Midterms are worth three times that, so midterms are worth 12%.

Random Homework Problems

Homework

Each time you load the page these problems change!

Review

Math 20 Topics

Here is a (not random) Math 20 review problem about geometry.

three carpet cleaning options

32. All the rooms in the floor plan below have ceilings 7' 6" high. The bedrooms and living room are carpeted. The bathroom, kitchen, and dining room have vinyl floors. (a) At $3 per square foot, what is the cost of replacing all the vinyl flooring? (b) Which of the three carpet cleaning services is the best buy, and how much will it cost? (c) If you need to paint the three non-closet walls of the first bedroom, what is the area that you will paint? (The balcony has a plastic "fake grass" surface. The closets are carpeted, but do not need cleaning. The T-shaped hallways has tile. Those areas are not part of our problem.)

two bedroom apartment floorplan

Icing on the Cake

Icing on a Gingerbread House

These homework problems have no answers. First do the other homework for as much practice as you need. Once you have confidence, these problems can be turned in to be graded.

33. When rolling two dice, what is the probability of the sum being 10 or more?

34. When rolling two dice, what is the probability of the sum being 5 or less?

35. When rolling two dice, what are the odds of the sum being 10 or more?

36. When rolling two dice, what are the odds of the sum being 5 or less?

37. Your friend is starting a food cart business. She has read that new food carts have a 35% chance to go out of business during the first year with a $10,000 loss, a 30% chance to earn $20,000 profit the first year, a 15% chance to earn $30,000 profit the first year, a 15% chance to earn $40,000 profit the first year, and a 5% chance to earn $50,000 profit the first year. Assuming these numbers are true, and your friend has typical skill and luck in her new business, what is the expected value of her first year's income?

Concept Tasks

cover image from Aha Insight book by Martin Gardener

These homework problems test your understanding in a deeper way. You may turn in written work or choose to share your work in front of the class. Be prepared to defend your understanding when your instructor asks you another question or two about your work.

38. The flu virus is spreading through a classroom of 30 students. 20 students have been vaccinated. There have been 9 cases of the flu. 3 students who had received the vaccine caught the flu. When measured as an absolute change (difference), how much did the vaccine reduce the chance of catching the flu? When measured as a relative change (percent change), how much does the vaccine reduce the chance of catching the flu?

39. A store's website allows people to rate how much they liked a certain purchase. 10 people rated it 0 stars, 24 rated it 1 star, 42 people rated it 2 stars, and 84 rated it 3 stars. What is the overall rating (the weighted average)?

40. There are three different routes for driving from home to work. The three routes are nearly the same distance, so it is only the traffic lights that matter for deciding which route is quickest. The first route crosses many small streets. There are five traffic lights, each has only a 15% chance of being red, and a red light delays you 1 minute. The second route crosses a few medium-sized streets. There are three traffic lights, each has a 40% chance of being red, and a red light delays you 2 minutes. The third route crosses one major street. There is only one traffic light, but it has a 75% chance of being red, and a red light delays you 3 minutes. Which route is expected to be quickest?

LCC Math 25 Packet Problems

Likelihood problems are in the packet on pages:
  • R-6 to R-12