MODULE 04 · PROBABILITY · LESSON 04.1
Probability Foundations — Continued
From basic probability properties to equally likely outcomes, multistage experiments, and independent practice.
Before you continue: two important ideas
In the finite models used here, where every listed elementary outcome has positive probability, only the empty event has probability 0, and only the whole sample space has probability 1.
Probability is not a short-run guarantee. A fair coin has probability 1/2 of heads, but two tosses could give HH, HT, TH, or TT. Probability does not prescribe an exact short sequence.
8. Fundamental Probability Properties
These six properties provide quick ways to test whether a probability calculation makes sense.
01 · Bounds
Every event has probability between 0 and 1, inclusive.
\(0 \leq P(A) \leq 1\)
Check: −0.2 and 1.3 are not valid probabilities.
02 · Whole Sample Space
The experiment must produce an outcome in its sample space.
\(P(\Omega)=1\)
03 · Empty Event
An impossible event contains no outcomes.
P(∅) = 0
04 · Elementary Probabilities Sum to One
For a finite sample space, the probabilities of all elementary outcomes add to 1.
\(\frac16+\frac16+\frac16+\frac16+\frac16+\frac16=1\)
This is the total for all six outcomes of a fair die.
05 · Add Probabilities Within an Event
If an event contains distinct elementary outcomes, add their probabilities. These exact outcomes cannot happen simultaneously in one roll.
\(P(\{1,2\})=\frac16+\frac16=\frac13\)
06 · Containment
If every outcome of A belongs to B, event A cannot be more likely than event B.
\(A\subseteq B \Rightarrow P(A)\leq P(B)\)
For a fair die:
\(A=\{6\},\quad B=\{4,5,6\}\)
\(P(A)=\frac16,\quad P(B)=\frac36=\frac12\)
Whenever A occurs, B also occurs.
9. Equally Likely Outcomes
If a finite sample space contains n equally likely outcomes, each has probability:
\(\frac{1}{n}\)
If event A contains m of these outcomes:
\(P(A)=\frac{m}{n}\)
Favorable outcomes ÷ total equally likely outcomes
Essential condition: The outcomes must actually be equally likely. Do not apply the formula simply because a list has n entries.
Worked Example 3 · Fair Die
Question: A fair die is rolled once. Find the probability of an even number.
\(\Omega=\{1,2,3,4,5,6\}\)
\(A=\{2,4,6\}\)
There are 6 equally likely outcomes and 3 favorable outcomes.
\(P(A)=\frac36=\frac12=50\%\)
Worked Example 4 · Numbered Cards
One card is selected uniformly from cards numbered 1 to 8. Find the probability of a number at least 6.
\(A=\{6,7,8\}\)
Three favorable outcomes among eight equally likely cards give:
\(P(A)=\frac38\)
Language check: “At least 6” includes 6.
10. Different Labels Do Not Mean Equal Probability
Worked Example 5 · Colored Balls
A bag contains 3 red balls and 1 blue ball. Every individual ball has the same chance of being selected.
| Individual outcomes | Recorded color |
|---|---|
| R₁, R₂, R₃ | Red |
| B₁ | Blue |
\(\Omega=\{R_1,R_2,R_3,B_1\}\)
\(R=\{R_1,R_2,R_3\}\)
\(P(\text{red})=\frac34,\quad P(\text{blue})=\frac14\)
Even if we record only {Red, Blue}, those two color outcomes are not equally likely.
11. Experiments With More Than One Stage
A complete outcome must describe the whole experiment.
Worked Example 6 · Two Coin Tosses
A coin is tossed twice, and order is recorded. The first letter means the first toss, the second letter means the second toss.
\(\Omega=\{HH,HT,TH,TT\}\)
Exactly one head:
\(A=\{HT,TH\}\)
| Outcome | Interpretation |
|---|---|
| HT | Heads first; tails second |
| TH | Tails first; heads second |
If only the number of heads is recorded: the values 0, 1, and 2 are possible, but they are not equally likely for two independent fair tosses.
Zero heads: TT. One head: HT or TH. Two heads: HH.
Key idea: A shorter list of recorded values does not necessarily represent equally likely outcomes. Systematic multistage calculations are developed separately.
12. Common Mistakes and Verification
01 · Outcome vs. event: “Even” is an event consisting of several exact-number outcomes.
02 · Incomplete sample space: {HH, HT, TT} leaves out TH.
03 · False equal-likelihood assumption: Two color labels do not imply two equally likely outcomes.
04 · “At least” confusion: “At least 4” includes 4.
05 · Probability above 1: A result such as 7/5 indicates an error.
06 · Short-run guarantee: Probability 1/2 does not force one occurrence in every two trials.
07 · Uncertainty vs. equality: Not knowing the outcome does not imply equal chances.
08 · Empty set vs. zero: ∅ has no outcomes; {0} has one outcome.
Verification Checklist
- Is the experiment clearly specified?
- Does the sample space include every possible outcome?
- Can exactly one listed elementary outcome occur per trial?
- Does the event contain precisely the required outcomes?
- Is equal likelihood stated or justified?
- Is the probability between 0 and 1?
- Do the elementary probabilities sum to 1?
13. Independent Practice
Try every question before opening the solutions below.
Practice 1 · Sample Space
A standard six-sided die is rolled once. Write the sample space.
Practice 2 · Prime Numbers
For the die in Practice 1, write the event A = “the result is prime.” Remember: 1 is not prime.
Practice 3 · Classify Events
For one standard die roll, classify each event as certain, impossible, or random:
- Result less than 7
- Result 9
- Result at least 5
Practice 4 · Two Coin Tosses
With order recorded, write the sample space, the event “both results are the same,” and the event “exactly one tail.”
Practice 5 · Valid Probabilities
Which values can be probabilities? −0.1, 0, 2/5, 1, 1.05, 75%
Practice 6 · Numbered Cards
A card is selected uniformly from 1 to 10. Write “greater than 7” as a set and find its probability.
Practice 7 · Unequal Color Groups
A bag contains 4 green balls and 2 yellow balls, each individual ball equally likely. Explain why green is not 1/2 and calculate its probability.
Practice 8 · Missing Probability
Three elementary outcomes have probabilities 0.2, q, and 0.5. Find q.
14. Practice Solutions
Open each answer only after attempting the corresponding question.
Solution 1 · Sample Space
\(\Omega=\{1,2,3,4,5,6\}\)
Solution 2 · Prime Numbers
\(A=\{2,3,5\}\)
The number 1 is not prime because it does not have exactly two positive divisors.
Solution 3 · Classify Events
- Certain — every die outcome is less than 7.
- Impossible — 9 is not on a standard die.
- Random — {5, 6} can occur or fail to occur.
Solution 4 · Two Coin Tosses
\(\Omega=\{HH,HT,TH,TT\}\)
Same results: \(A=\{HH,TT\}\)
Exactly one tail: \(B=\{HT,TH\}\)
Solution 5 · Valid Probabilities
Valid: 0, 2/5, 1, 75% (= 0.75).
Invalid: −0.1, 1.05. All probabilities must be between 0 and 1.
Solution 6 · Numbered Cards
\(A=\{8,9,10\}\)
\(P(A)=\frac{3}{10}\)
Solution 7 · Unequal Color Groups
There are 6 equally likely balls, of which 4 are green. The two colors are not equally likely.
\(P(\text{green})=\frac46=\frac23\)
Solution 8 · Missing Probability
\(0.2+q+0.5=1\)
\(q=1-0.2-0.5=0.3\)
This satisfies the probability bounds.
15. Lesson Self-Check
This is a short formative check within 04.1, not a replacement for the scheduled probability quiz. Select one best answer per question.
Check 1
Which set represents “at most 2” for one die roll?
A. {1}
B. {1, 2}
C. {2, 3, 4, 5, 6}
D. {3, 4, 5, 6}
Check 2
Which statement is always true?
A. Every uncertain experiment has equally likely outcomes.
B. A probability may exceed 1 if the event is very likely.
C. Every event contains exactly one outcome.
D. Every event has probability between 0 and 1, inclusive.
Check 3
A bag has five red balls and one blue ball; each ball is equally likely. What is P(blue)?
A. 1/6
B. 1/2
C. 5/6
D. 1/5
Check 4
With two coin tosses and order recorded, which event means “exactly one head”?
A. {HH}
B. {HH, HT, TH}
C. {HT, TH}
D. {TT}
Check 5
A student says: “Heads has probability 1/2, so two fair coin tosses must contain exactly one head.” Which response is correct?
A. Correct: probabilities fix every two-toss sequence.
B. Incorrect: HH and TT are also possible.
C. Correct, but only if the first toss is tails.
D. Incorrect: the probability of at least one head in two independent fair tosses is 1.
16. Self-Check Answers
| Check | Correct | Why |
|---|---|---|
| 1 | B | “At most 2” includes 1 and 2. |
| 2 | D | Every probability lies from 0 to 1, inclusive. |
| 3 | A | One blue ball out of six equally likely balls. |
| 4 | C | Exactly one head: HT or TH. |
| 5 | B | HH and TT remain possible. D is false because TT remains possible. |
17. Lesson Review
You should now be able to:
- Describe a random experiment and distinguish outcomes from events.
- List a complete sample space and represent events as sets.
- Identify certain, impossible, and random events.
- Interpret “exactly,” “at least,” and “at most.”
- Read and write probability notation as fractions, decimals, and percentages.
- Apply the fundamental probability bounds.
- Recognize when equal likelihood is justified.
- Explain why probability is not a short-run guarantee.
Essential Formulas
\(0\leq P(A)\leq1\)
\(P(\Omega)=1\)
P(∅) = 0
\(P(A)=\frac{\text{favorable outcomes}}{\text{total equally likely outcomes}}\)
Completion target: Describe a sample space and event correctly, explain the equal-likelihood condition, and independently correct every missed practice or self-check question.