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Lesson 7 of 7

The Mathematics of Maybe: Introduction to Probability · Lesson 7 of 7

Chapter Summary and Practice

“Take one final chance on the chapter—fortunately, practice improves the odds.”

Learning Objectives

• Recall the main language and formulas of probability. • Distinguish experimental and theoretical probability. • Construct sample spaces and events. • Interpret high probabilities without confusing them with certainty. • Use tree diagrams and long-run reasoning in mixed problems.

Probability gives us a mathematical language for uncertainty. It does not remove randomness; instead, it helps us measure likelihood, organise possible outcomes and use evidence or fair-outcome reasoning to make sensible estimates.

Probability and Randomness

A random experiment has known possible outcomes but an unpredictable individual result. Probability measures how likely an event is.

Probability rangeLaTeX

Probability Scale and Certainty

P=0 means impossible and P=1 means certain. Values between 0 and 1 indicate different degrees of likelihood. A probability such as 0.99 means very likely, not guaranteed, because 0.99 is still less than 1.

Remember the 99% idea

If 99 out of 100 equally likely outcomes are wins, one losing outcome still exists. That one outcome may occur on the next trial. Only 100 out of 100 favourable outcomes would make the event certain.

Experimental Probability

Experimental probabilityLaTeX

Experimental probability is based on actual data and may vary when the number of trials is small.

Theoretical Probability

Theoretical probability for equally likely outcomesLaTeX

This formula is appropriate when the outcomes are equally likely, as with an ideal fair coin or fair die.

Sample Spaces and Events

The sample space S contains every possible outcome exactly once. An event E is a subset of S. The number of elements in S is written n(S).

Long-Run Behaviour

As the number of fair trials increases, experimental probability tends to get closer to theoretical probability. But this does not mean the very next trial must correct a streak. Independent random trials have no memory.

Tree Diagrams

Tree diagrams display multi-step experiments one stage at a time. Each full path corresponds to one complete outcome.

How to Choose the Right Method

SituationUseful idea
Observed repeated trialsExperimental probability
Fair equally likely outcomesTheoretical probability
Survey or past dataStatistical estimate / relative frequency
Need all possible outcomesSample space
Need selected outcomes from SEvent
Multi-step experimentTree diagram
Long sequence of independent fair trialsLaw of Large Numbers; avoid Gambler's Fallacy
Common Mistakes

• Treating a high probability as a guarantee, • Calling a rare event impossible even when its probability is greater than 0, • Using favourable/total without checking that outcomes are equally likely, • Leaving outcomes out of a sample space or listing one twice, • Confusing an event with a single outcome, • Assuming a coin or die 'remembers' previous results, • Expecting experimental probability to equal theoretical probability exactly after only a few trials,

Guided Practice

Guided Example 1: Probability Scale

Problem
A bag contains 20 identical slips, 19 marked YES and 1 marked NO. Find P(YES). Is YES certain?

  1. 1.P(YES)=19/20=0.95=95%.
  2. 2.0.95<1.
  3. 3.Therefore YES is very likely but not certain.
  4. 4.The NO slip still makes the opposite outcome possible.
Guided Example 2: Experimental Probability

Problem
A spinner is used 50 times and lands on green 18 times. Find the experimental probability of green.

  1. 1.Pexperimental(green)=18/50.
  2. 2.18/50=0.36=36%.
Guided Example 3: Theoretical Probability

Problem
A fair die is rolled. Find the probability of getting a number greater than 4.

  1. 1.S={1,2,3,4,5,6}.
  2. 2.Event E={5,6}.
  3. 3.P(E)=2/6=1/3.
Guided Example 4: Two Coins

Problem
Two fair coins are tossed. Find the probability of at least one Head.

  1. 1.S={HH,HT,TH,TT}.
  2. 2.E={HH,HT,TH}.
  3. 3.P(E)=3/4.
Guided Example 5: Gambler's Fallacy

Problem
A fair die has shown 6 four times in a row. What is the probability of 6 on the next roll?

  1. 1.The next roll is independent of the previous rolls.
  2. 2.A fair die has one 6 among six equally likely faces.
  3. 3.P(6)=1/6.
  4. 4.The streak does not change that probability.

Practice Problems

Practice Questions
  1. Rank these events from impossible to certain and explain: rolling 9 on a standard die; getting Heads on a fair coin; choosing a red sweet from a bag containing only red sweets; seeing rain tomorrow.
  2. A bag contains 40 sweets: 14 red, 11 green, 9 yellow and 6 blue. Using the observed composition, find the probability of selecting green.
  3. A school survey asks 50 students about clubs; 17 choose Science, 13 Arts, 12 Sports and 8 Debate. Estimate the probability a selected student prefers Sports, then estimate how many out of 600 students might prefer Sports.
  4. Toss a coin 30 times and calculate the experimental probability of Heads. Compare it with 1/2.
  5. A fair die is rolled 18 times and a 3 appears 5 times. Calculate experimental and theoretical probabilities of rolling 3.
  6. Write the sample space for rolling a die and tossing a coin together.
  7. A box contains 4 green and 6 red balls. One ball is drawn and only its colour is recorded. Write the sample space.
  8. At a fair there are 3 snacks and 2 drinks. List all snack-drink outcomes.
  9. Two coins are tossed. Find the probability of exactly one Head.
  10. Ten cards numbered 1 to 10 are mixed. Find the probability of drawing an even number.
  11. A bag contains 4 red, 3 blue and 2 green balls. Find the probability that one randomly selected ball is not red.
  12. Cards spell PEACE. One card is selected at random. Find P(E) and P(not E).
  13. Draw a tree diagram for selecting one fruit from a basket containing Apple or Orange and then one drink from Chai or Lassi.
  14. A fair coin lands Heads seven times in a row. Explain why P(Tails on the next toss)=1/2.
  15. A machine succeeds 99 times out of 100 in a long-run model. Explain carefully why a single operation is still not guaranteed to succeed.

Quiz

Quick check

Which probability represents certainty?

Quick check

A fair coin shows Heads six times in a row. What is P(Tails) on the next toss?

Quick check

Which formula describes experimental probability?

Quick check

For two coin tosses, which is the complete sample space?

Quick check

What does the Law of Large Numbers suggest?

Before You Finish the Chapter

Check that you can explain randomness in everyday language, use the 0-to-1 probability scale, distinguish 99% from certainty, calculate experimental and theoretical probability, construct sample spaces and events, interpret sample data, explain the Law of Large Numbers, avoid Gambler's Fallacy, and use tree diagrams for multi-step experiments.

Key Takeaways

Key Takeaways

• Probability measures uncertainty rather than predicting one outcome with certainty. • Only P=1 is guaranteed; 99% is still not certain. • Experimental probability uses observed data. • Theoretical probability uses equally likely outcomes. • Sample spaces organise all possibilities; events select outcomes of interest. • Tree diagrams organise multi-step experiments. • Long-run patterns do not make independent trials remember the past.

Coming Next

This completes the introduction to probability. The most important habit to carry forward is to separate 'likely' from 'certain' and to describe the possible outcomes before calculating.

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Tree Diagrams

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