A. Counting and Probability · Probability

Probability as a Measure of Likelihood

A1.1About 65 minutes

Learning goals

I can describe probability as a number from 0 to 1 that measures how likely an event is.
I can write a probability as a fraction, a decimal and a percent, and place it on the probability scale.
I can tell theoretical, experimental and subjective probability apart.
I can interpret probabilities in real situations, such as weather forecasts and product quality control.

Words to know

probability
A number from 0 to 1 that measures how likely an event is. 0 means impossible and 1 means certain.

experiment
Any process with an uncertain result that can be repeated, such as rolling a die or testing a light bulb.

outcome
One possible result of an experiment.

event
A set of one or more outcomes, such as "rolling an even number".

theoretical probability
A probability found by reasoning about equally likely outcomes, without doing the experiment.

experimental probability
A probability estimated from data: the number of times the event happened divided by the number of trials. Also called empirical probability or relative frequency.

subjective probability
A probability based on judgment, experience or opinion rather than on calculation or data.

Welcome to Data Management

Every day you hear numbers that measure uncertainty. "There is a 60% chance of snow tomorrow." "About 1 in 250 of these phones is returned for a fault." "Our team has a good shot at the playoffs." Probability is the branch of mathematics that turns statements like these into numbers you can calculate with, compare and test.

This course, Mathematics of Data Management (MDM4U), has two big halves: probability (this unit, then Permutations and Combinations, Discrete Probability Distributions and Continuous Distributions and the Normal Distribution) and statistics (organizing, analysing and judging data). Probability is the foundation of statistics, so we start here.

Remember from Grades 7 and 8: you compared theoretical and experimental probability, used tree diagrams for independent and dependent events, and used Venn diagrams to find P(A or B). In this unit you will make all of those ideas precise, with proper notation and formulas, and use them on harder problems.

The probability scale

The probability of an event A is written P(A). It is always a number from 0 to 1:

P(A) = 0 means A is impossible (rolling a 7 on a standard die).
P(A) = 1 means A is certain (rolling a number from 1 to 6).
Values in between measure how likely A is. The closer to 1, the more likely.

So 0 ≤ P(A) ≤ 1 for every event. A probability of 1.3 or −0.2 is always a mistake.

You can write the same probability three ways: as a fraction (1/4), a decimal (0.25) or a percent (25%). Fractions are exact; decimals and percents are easier to compare.

The probability scale runs from 0 (impossible) to 1 (certain). The events use one fair die or one coin. Each dot is an event placed at its probability: rolling a 6 is 1/6 ≈ 0.17, a fair coin landing heads is 0.5, and a 70% chance of precipitation is 0.7.

Three ways to get a probability

Theoretical probability comes from reasoning. If a die is fair, its 6 faces are equally likely, so P(roll a 6) = 1/6. You do not need to roll it at all. You will calculate these carefully in lesson 3.
Experimental (empirical) probability comes from data. Roll a die 600 times and count the sixes: if you get 94 sixes, the experimental probability is 94/600 ≈ 0.157. This is the only kind available when outcomes are not equally likely, such as a thumbtack landing point-up or a new medicine working.
Subjective probability comes from judgment. A coach who says "I'd put our chances of winning at about 70%" is giving a subjective probability. It may be well informed, but it is an estimate, not a calculation.

The long-run meaning connects them all: P(A) = 0.3 means that if the experiment were repeated a very large number of times, A would happen in about 30% of the trials. It does not promise that A happens exactly 3 times in every 10.

Worked example

Example 1: Three forms and the scale

A survey company estimates that the probability a randomly chosen adult in a town owns a snowblower is 0.38. Write 0.38 as a fraction and a percent, and describe the likelihood in words. Then decide which is more likely: an event with probability 3/8 or one with probability 0.38.

0.38 = 38/100 = 19/50, and as a percent it is 38%.
0.38 is below 0.5, so the event is less likely than not, but far from impossible.
Convert 3/8 to a decimal: 3 ÷ 8 = 0.375.
Compare: 0.375 < 0.38.

0.38 = 19/50 = 38%. The event with probability 0.38 is (very slightly) more likely than the one with 3/8 = 0.375.

Probability in the real world

Weather. Environment and Climate Change Canada forecasts give a probability of precipitation (POP). A POP of 40% means there is a 40% chance that measurable rain or snow will fall at a given spot in the forecast region during the forecast period. It does not mean it will rain for 40% of the day or over 40% of the region.

Quality control. A factory that makes light bulbs tests samples. If about 4 in every 1000 bulbs are defective, then P(a bulb is defective) ≈ 0.004, and in a box of 500 bulbs it expects about 0.004 × 500 = 2 defective bulbs. The actual number in one box might be 0, 1, 3 or more; 2 is the long-run average.

Insurance and health. Actuaries (careers that need strong probability and statistics) set insurance prices from the probabilities of accidents, fires or floods. Public health agencies use probabilities to describe the risk of side effects or the chance a test is accurate.

If P(defective) = 0.004, a batch of 500 bulbs is expected to contain about 0.004 × 500 = 2 defective bulbs. Another batch might have 0, 1 or 4: the expected number is a long-run average, not a promise.

Worked example

Example 2: Which kind of probability?

Classify each probability as theoretical, experimental or subjective.

(a) A fair spinner has 8 equal sections, 3 of them blue. P(blue) = 3/8.

(b) Of 1250 drivers checked, 1180 were wearing seatbelts, so P(wearing a seatbelt) ≈ 0.944.

(d) A basketball player has made 168 of her last 210 free throws, so P(make the next one) ≈ 0.8.

(a) It comes from reasoning about equally likely sections, with no experiment: theoretical.
(b) It comes from counting results in observed data: experimental.
(c) It is a personal judgment: subjective.
(d) It is based on 210 recorded attempts: experimental. 168 ÷ 210 = 0.8.

(a) theoretical, (b) experimental, (c) subjective, (d) experimental.

Worked example

Example 3: Using a probability to predict

A phone maker finds that P(a phone is returned under warranty) = 0.012. A store in Sudbury sells 750 of these phones in a year.

(a) About how many returns should the store expect?

(b) Does this mean exactly that many phones will be returned?

(a) Multiply the probability by the number of trials: 0.012 × 750 = 9.
(b) No. 9 is the expected (long-run average) number. The real count will vary from year to year and from store to store, for example 6 one year and 12 the next. Over many stores and years, the average gets close to 9 per 750 phones.

(a) About 9 returns. (b) No, 9 is an expected value, not a guarantee. (Expected value gets a full treatment in the unit Discrete Probability Distributions.)

Two classic misunderstandings

"It's 50-50: either it happens or it doesn't." Two possible outcomes do not make them equally likely. Either you win the lottery or you don't, but those outcomes are nowhere near 50-50.
"It's due." If a fair coin lands heads 5 times in a row, P(heads) on the next toss is still 1/2. The coin has no memory. Believing otherwise is called the gambler's fallacy.

Just for fun

Weather forecasting may be the rare job where you can look wrong 30% of the time and still be perfectly right: if it stays dry on 3 out of every 10 days with a 70% POP (and rains on the other 7), the forecasts were spot on.

Hands-on activity

The bottle cap experiment

You need: a plastic bottle cap (or a paper cup), a flat table, and paper to record results.

Before you start, write down your subjective probability that the cap lands open side up when dropped from about 30 cm.
Is there a theoretical probability? Explain why not. (Are the two ways of landing equally likely?)
Drop the cap 50 times. Record U (open side up) or D (open side down) in rows of 10.
After every 10 drops, calculate the experimental probability so far: (number of U) ÷ (drops so far).
Compare your final experimental probability with your subjective guess.

What to look for: there is no theoretical probability because the cap's two sides are not shaped the same, so the outcomes are not equally likely. Your experimental probability will settle down as the number of drops grows. Results are often far from 50%, but which side wins depends on your cap and table: the data decide. Keep your results; you will use the same idea in lesson 5.

Your turn

Write 0.65 as a fraction in lowest terms and as a percent.

Which is more likely: an event with probability 5/12 or one with probability 0.4?

Explain why P(A) = 1.25 is impossible.

A fair die is rolled. Describe in words and give the probability of (a) rolling a 9, (b) rolling a number less than 7.

Classify: "Based on 40 years of its own weather records, a town finds the probability of a white Christmas there is about 0.8."

The probability that a battery from a certain batch is dead is 0.02. How many dead batteries would you expect in a package of 24? In a shipment of 5000?

A forecast says POP 30%. Your friend says, "So it will rain for about 30% of the day." Correct this.

More practice: warm-up

Write 3/4 as a decimal and a percent.

What is the probability of an impossible event?

What is the probability of a certain event?

Write 8% as a decimal.

A coin has been flipped 4 times and landed tails each time. What is P(heads) on the next flip of this fair coin?

More practice: on your own

Put these in order from least to most likely: 0.3, 2/9, 35%, 1/3, 0.29.

In a survey of 400 Grade 12 students, 92 said they walk to school. Estimate the probability that a randomly chosen student from this group walks to school. What kind of probability is this?

A goalie has stopped 456 of the last 500 shots she faced. Estimate the probability she stops the next shot, and how many goals she might allow on the next 30 shots.

Explain the gambler's fallacy using a roulette-style spinner that has landed on red 6 times in a row.

A new medicine helped 340 of 400 patients in a trial. Why can't we find a theoretical probability that it helps the next patient?

Challenge

A machine produces bolts. P(defective) = 0.003. A quality inspector says, "We made 2000 bolts today, so exactly 6 are defective." Explain the error and give a better statement.

Two forecasters each issued many forecasts of "POP 60%". On those days, measurable precipitation fell on 59% of Forecaster A's days and 82% of Forecaster B's days. Which forecaster is better calibrated? Explain.

A website claims, "The probability you'll love this movie is 97%." Is this theoretical, experimental or subjective? What information would you want before trusting it?

Check your understanding

Quiz

Which value could not be a probability?

A spinner has 5 equal sections numbered 1 to 5. The probability of spinning a 3, found by reasoning about the sections, is an example of:

A forecast gives a POP of 70% for Thunder Bay tomorrow. Which statement is correct?

P(a component is faulty) = 0.005. About how many faulty components are expected in 2400 components?

A fair coin has landed heads 7 times in a row. What is the probability it lands heads on the next toss?

Which is the most likely event?

For parents and teachers

Big idea (A1.1): a probability is a number from 0 to 1 that measures how likely something is. It can come from reasoning (theoretical), from data (experimental) or from judgment (subjective), and in every case it describes what happens in the long run, not in one try.

How to help: Listen for probability language together: weather forecasts, sports commentary, product warranties. Ask your teen, "What kind of probability is that? Where would the number come from?" Ask them to convert a fraction such as 3/8 to a decimal and a percent.

Common difficulties:

Treating two possible outcomes as automatically 50-50.
The gambler's fallacy ("it's due").
Reading an expected number (0.004 × 500 = 2) as a guarantee.
Misreading POP as the share of the day or of the region that gets rain.

The bottle cap activity is quick and works well as a family challenge: everyone guesses first, then the data decide.