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Probability — Assignment Prep

Basic probability, letters, conditions, cards, OR, NOT, and experimental vs theoretical — Concept, Worked Example, and Practice for each skill.

F1 — What Is Probability?

Probability measures how likely an event is.It is always between (impossible) and (certain). You can write it as a fraction, a decimal, or a percentage.Key idea: favourable ÷ total — then simplify the fraction if you can.

Example 1 — Worked example — fair coin

A fair coin has 2 equally likely outcomes: heads or tails. P (heads) = \frac{1}{2} = 0.5 = 50%

F2 — Probability with Letters in a Word

When a letter is chosen at random from a word, letters that appear more often are more likely. Key idea: count the total letters, count each distinct letter, then \frac{count of that letter}{total} . The most likely letter is simply the one with the highest count.

Example 1 — Worked example — CHEESE

Word: CHEESE (6 letters). Counts: C = 1, H = 1, E = 3, S = 1 . Most likely letter: E P (E) = \frac{3}{6} = \frac{1}{2}

Practice 1 . From the word BANANA, which letter is most likely, and what is its probability? Answer: A appears 3 times out of 6 \to most likely A, P (A) = \frac{3}{6} = \frac{1}{2} .

Practice 2 . From the word APPLE, which letter is most likely, and what is its probability? Answer: P appears 2 times out of 5 \to most likely P, P (P) = \frac{2}{5} .

F3 — Probability with a Condition

To find P (even), P (odd), or P (greater than k) : 1 Check every value against the condition, one at a time. 2 Count how many satisfy it. 3 Put that count over the total count. Key idea: favourable = "passes the test"; total = size of the set.

Example 1 — Worked example — P(even)

Set: 4, 15, 22, 7, 30, 11, 18, 25 Evens: 4, 22, 30, 18 \to 4 values. Total: 8 values. P (even) = \frac{4}{8} = \frac{1}{2}

Practice 1 . From 3, 8, 12, 5, 9, 14, 7, find P (even) . Answer: Evens 8, 12, 14 \to \frac{3}{7} .

Practice 2 . From 21, 16, 33, 10, 25, 18, 27, 12, find P (greater than 20) . Answer: Values > 20 : 21, 33, 25, 27 \to \frac{4}{8} = \frac{1}{2} .

F4 — Probability with a Standard Deck of Cards

A standard deck has: 52 cards in total 4 suits: hearts, diamonds, clubs, spades (13 ranks each) 26 red and 26 black 12 face cards (Jack, Queen, King — 3 per suit) Honor cards (as used in many exams) = Ace, Jack, Queen, King \to 4 per suit, 16 in the deck Key idea: memorise the deck structure first; then every card probability is just a count over 52 .

Example 1 — Worked examples — King and red

P (King) = \frac{4}{52} = \frac{1}{13} P (red card) = \frac{26}{52} = \frac{1}{2}

Practice 1 . Find P (hearts) . Answer: \frac{13}{52} = \frac{1}{4} .

Practice 2 . Find P (face card) . Answer: 12 face cards \to \frac{12}{52} = \frac{3}{13} .

F5 — Combining Probabilities — the OR Rule

If two outcomes cannot happen at the same time (mutually exclusive), then: P (A or B) = P (A) + P (B) Key idea: "or" of mutually exclusive events \to add .

Example 1 — Worked example — bag of counters

A bag holds 40 counters: 15 red, 10 blue, 8 green, 7 yellow. A counter cannot be two colours at once, so: P (red or blue) = \frac{15}{40} + \frac{10}{40} = \frac{25}{40} = \frac{5}{8}

Practice 1 . Same bag. Find P (green or yellow) . Answer: \frac{8}{40} + \frac{7}{40} = \frac{15}{40} = \frac{3}{8} .

Practice 2 . A bag of 30 counters: 12 red, 8 blue, 6 green, 4 yellow. Find P (red or green) . Answer: \frac{12}{30} + \frac{6}{30} = \frac{18}{30} = \frac{3}{5} .

F6 — The Complement Rule — NOT an Event

The complement of an event A is "not A". P (not A) = 1 - P (A) You can also count everything that is not A directly — both routes must agree. Key idea: "not A" = everything else = 1 - P (A) .

Example 1 — Worked example — not green

Same 40 -counter bag (8 green). Shortcut: P (not green) = 1 - \frac{8}{40} = 1 - 0.2 = 0.8 Direct count: Not green = 15 + 10 + 7 = 32 counters \to \frac{32}{40} = 0.8 Both routes match.

Practice 1 . Same 40 -counter bag. Find P (not red) . Answer: 1 - \frac{15}{40} = \frac{25}{40} = \frac{5}{8} (or 10 + 8 + 7 = 25 over 40).

Practice 2 . The 30 -counter bag (12 red, 8 blue, 6 green, 4 yellow). Find P (not yellow) . Answer: 1 - \frac{4}{30} = \frac{26}{30} = \frac{13}{15} .

F7 — Experimental vs Theoretical Probability

Theoretical probability comes from the structure of the situation *before* anything happens (equally likely outcomes). Experimental (empirical) probability comes from what was *actually observed*: P_{experimental} (event) = \frac{number of times the event happened}{number of trials actually run} Key idea: divide by the number of trials you ran, not by some other total sitting in the background (for example, a container of 500 marbles does not change a 200 -trial experiment — every experimental probability is still "out of 200 ").

Example 1 — Worked example — die rolled 60 times

Results: Outcome 123456 Times rolled 8119121010 Check: 8 + 11 + 9 + 12 + 10 + 10 = 60 ✓ P_{experimental} (rolling a 4) = \frac{12}{60} = 0.2 P_{theoretical} (rolling a 4) = \frac{1}{6} \approx 0.167 They are close, but not identical — that is normal for a finite experiment.

Outcome	$1$	$2$	$3$	$4$	$5$	$6$
Times rolled	$8$	$11$	$9$	$12$	$10$	$10$

Die rolled 60 times
Die rolled 60 times

Practice 1 . A fair 4 -section spinner is spun 40 times: Section 1234 Times 791410 Find the experimental probability of landing on 3 . Answer: \frac{14}{40} = 0.35 (divide by 40 trials, not by anything else).

Section	$1$	$2$	$3$	$4$
Times	$7$	$9$	$14$	$10$

Practice 2 . Using the same spinner results, state the theoretical probability of landing on 3 (assume each section is equally likely), and compare it with the experimental value from Practice 1 . Answer: Theoretical P (3) = \frac{1}{4} = 0.25 . Experimental was 0.35 — different because the experiment is finite; with more spins, experimental usually moves closer to theoretical.

More lessons in Basic Probability · Next: Introduction to Probability

F1 — What Is Probability?

Probability measures how likely an event is.It is always between (impossible) and (certain). You can write it as a fraction, a decimal, or a percentage.Key idea: favourable ÷ total — then simplify the fraction if you can.

Example 1 — Worked example — fair coin

A fair coin has 2 equally likely outcomes: heads or tails. P (heads) = \frac{1}{2} = 0.5 = 50%

F2 — Probability with Letters in a Word

When a letter is chosen at random from a word, letters that appear more often are more likely. Key idea: count the total letters, count each distinct letter, then \frac{count of that letter}{total} . The most likely letter is simply the one with the highest count.

Example 1 — Worked example — CHEESE

Word: CHEESE (6 letters). Counts: C = 1, H = 1, E = 3, S = 1 . Most likely letter: E P (E) = \frac{3}{6} = \frac{1}{2}

Practice 1 . From the word BANANA, which letter is most likely, and what is its probability? Answer: A appears 3 times out of 6 \to most likely A, P (A) = \frac{3}{6} = \frac{1}{2} .

Practice 2 . From the word APPLE, which letter is most likely, and what is its probability? Answer: P appears 2 times out of 5 \to most likely P, P (P) = \frac{2}{5} .

F3 — Probability with a Condition

To find P (even), P (odd), or P (greater than k) : 1 Check every value against the condition, one at a time. 2 Count how many satisfy it. 3 Put that count over the total count. Key idea: favourable = "passes the test"; total = size of the set.

Example 1 — Worked example — P(even)

Set: 4, 15, 22, 7, 30, 11, 18, 25 Evens: 4, 22, 30, 18 \to 4 values. Total: 8 values. P (even) = \frac{4}{8} = \frac{1}{2}

Practice 1 . From 3, 8, 12, 5, 9, 14, 7, find P (even) . Answer: Evens 8, 12, 14 \to \frac{3}{7} .

Practice 2 . From 21, 16, 33, 10, 25, 18, 27, 12, find P (greater than 20) . Answer: Values > 20 : 21, 33, 25, 27 \to \frac{4}{8} = \frac{1}{2} .

F4 — Probability with a Standard Deck of Cards

A standard deck has: 52 cards in total 4 suits: hearts, diamonds, clubs, spades (13 ranks each) 26 red and 26 black 12 face cards (Jack, Queen, King — 3 per suit) Honor cards (as used in many exams) = Ace, Jack, Queen, King \to 4 per suit, 16 in the deck Key idea: memorise the deck structure first; then every card probability is just a count over 52 .

Example 1 — Worked examples — King and red

P (King) = \frac{4}{52} = \frac{1}{13} P (red card) = \frac{26}{52} = \frac{1}{2}

Practice 1 . Find P (hearts) . Answer: \frac{13}{52} = \frac{1}{4} .

Practice 2 . Find P (face card) . Answer: 12 face cards \to \frac{12}{52} = \frac{3}{13} .

F5 — Combining Probabilities — the OR Rule

If two outcomes cannot happen at the same time (mutually exclusive), then: P (A or B) = P (A) + P (B) Key idea: "or" of mutually exclusive events \to add .

Example 1 — Worked example — bag of counters

A bag holds 40 counters: 15 red, 10 blue, 8 green, 7 yellow. A counter cannot be two colours at once, so: P (red or blue) = \frac{15}{40} + \frac{10}{40} = \frac{25}{40} = \frac{5}{8}

Practice 1 . Same bag. Find P (green or yellow) . Answer: \frac{8}{40} + \frac{7}{40} = \frac{15}{40} = \frac{3}{8} .

Practice 2 . A bag of 30 counters: 12 red, 8 blue, 6 green, 4 yellow. Find P (red or green) . Answer: \frac{12}{30} + \frac{6}{30} = \frac{18}{30} = \frac{3}{5} .

F6 — The Complement Rule — NOT an Event

The complement of an event A is "not A". P (not A) = 1 - P (A) You can also count everything that is not A directly — both routes must agree. Key idea: "not A" = everything else = 1 - P (A) .

Example 1 — Worked example — not green

Same 40 -counter bag (8 green). Shortcut: P (not green) = 1 - \frac{8}{40} = 1 - 0.2 = 0.8 Direct count: Not green = 15 + 10 + 7 = 32 counters \to \frac{32}{40} = 0.8 Both routes match.

Practice 1 . Same 40 -counter bag. Find P (not red) . Answer: 1 - \frac{15}{40} = \frac{25}{40} = \frac{5}{8} (or 10 + 8 + 7 = 25 over 40).

Practice 2 . The 30 -counter bag (12 red, 8 blue, 6 green, 4 yellow). Find P (not yellow) . Answer: 1 - \frac{4}{30} = \frac{26}{30} = \frac{13}{15} .

F7 — Experimental vs Theoretical Probability

Theoretical probability comes from the structure of the situation *before* anything happens (equally likely outcomes). Experimental (empirical) probability comes from what was *actually observed*: P_{experimental} (event) = \frac{number of times the event happened}{number of trials actually run} Key idea: divide by the number of trials you ran, not by some other total sitting in the background (for example, a container of 500 marbles does not change a 200 -trial experiment — every experimental probability is still "out of 200 ").

Example 1 — Worked example — die rolled 60 times

Results: Outcome 123456 Times rolled 8119121010 Check: 8 + 11 + 9 + 12 + 10 + 10 = 60 ✓ P_{experimental} (rolling a 4) = \frac{12}{60} = 0.2 P_{theoretical} (rolling a 4) = \frac{1}{6} \approx 0.167 They are close, but not identical — that is normal for a finite experiment.

Outcome	$1$	$2$	$3$	$4$	$5$	$6$
Times rolled	$8$	$11$	$9$	$12$	$10$	$10$

Die rolled 60 times
Die rolled 60 times

Practice 1 . A fair 4 -section spinner is spun 40 times: Section 1234 Times 791410 Find the experimental probability of landing on 3 . Answer: \frac{14}{40} = 0.35 (divide by 40 trials, not by anything else).

Section	$1$	$2$	$3$	$4$
Times	$7$	$9$	$14$	$10$

Practice 2 . Using the same spinner results, state the theoretical probability of landing on 3 (assume each section is equally likely), and compare it with the experimental value from Practice 1 . Answer: Theoretical P (3) = \frac{1}{4} = 0.25 . Experimental was 0.35 — different because the experiment is finite; with more spins, experimental usually moves closer to theoretical.