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Recap F — Probability

The probability formula, dice, letters, numbers, a pack of cards and experimental probability — 12 worked examples.

The Big Idea

Probability measures how likely something is, on a scale from 0 (impossible) to 1 (certain). An even chance is \frac{1}{2} .

1. Dice, Letters and Numbers

Example 1 — a die

Question: A fair six-sided die is rolled. Find P (6) and P (even) . Answer: Outcomes: 1, 2, 3, 4, 5, 6 (6 in total). P (6) = \frac{1}{6} Even: 2, 4, 6 \to P (even) = \frac{3}{6} = \frac{1}{2}

Example 2 — not

Question: Using the same die, find P (not 6) . Answer: P (not 6) = 1 - \frac{1}{6} = \frac{5}{6} (the five outcomes 1 - 5)

Example 3 — the most likely letter

Question: A letter is picked at random from BUBBLE . Which letter is most likely? Answer: Count each letter: B = 3, U = 1, L = 1, E = 1 (6 letters). P (B) = \frac{3}{6} = \frac{1}{2}, the highest — B is most likely.

Example 4 — vowels

Question: A letter is picked at random from ORANGE . Find P (vowel) . Answer: Vowels: O, A, E \to 3 out of 6 . P (vowel) = \frac{3}{6} = \frac{1}{2}

Example 5 — even numbers in a list

Question: A number is picked from 5, 8, 11, 14, 20, 3 . Find P (even) . Answer: Even: 8, 14, 20 \to 3 out of 6 . P (even) = \frac{3}{6} = \frac{1}{2}

Example 6 — multiples

Question: A whole number from 1 to 30 is chosen. Find P (multiple of 5) . Answer: Multiples of 5 : 5, 10, 15, 20, 25, 30 \to 6 out of 30 . P = \frac{6}{30} = \frac{1}{5}

2. A Pack of Cards

Tip

Key facts — 52 cards

Suit Colour Cards Hearts ♥ red A, 2 - 10, J, Q, K (13 cards) Diamonds ♦ red 13 cards Clubs ♣ black 13 cards Spades ♠ black 13 cards So there are 4 of each value (4 aces, 4 kings \dots), 26 red and 26 black cards, and 12 picture cards (J, Q, K). Honour cards are A, J, Q, K — 16 in total.

Example 1 — one value

Question: Find P (queen) . Answer: 4 queens: \frac{4}{52} = \frac{1}{13}

Example 2 — a colour or a suit

Question: Find P (black card) and P (diamond) . Answer: Black: \frac{26}{52} = \frac{1}{2} Diamond: \frac{13}{52} = \frac{1}{4}

Example 3 — picture and honour cards

Question: Find P (picture card) and P (honour card: A, J, Q or K) . Answer: Picture (J, Q, K): 3 \times 4 = 12 \to \frac{12}{52} = \frac{3}{13} Honour (A, J, Q, K): 4 \times 4 = 16 \to \frac{16}{52} = \frac{4}{13}

Example 4 — or

Question: Find P (ace or king) and P (red king) . Answer: Ace or king: 4 + 4 = 8 \to \frac{8}{52} = \frac{2}{13} Red king (hearts and diamonds): \frac{2}{52} = \frac{1}{26}

3. Experimental Probability

Example 1 — a spinner

Question: A spinner is spun 50 times : red 20, blue 18, green 12 . Find P (red), P (red or blue) and P (not green) . Answer: P (red) = \frac{20}{50} = \frac{2}{5} = 0.4 P (red or blue) = \frac{20 + 18}{50} = \frac{38}{50} = 0.76 — add the frequencies P (not green) = 1 - \frac{12}{50} = \frac{38}{50} = 0.76

Example 2 — a bag of marbles (like the paper)

Question: A bag holds 300 marbles of three colours. A marble is drawn and replaced, 120 times: red 54, blue 42, green 24 . Find P (red), P (blue or green), P (not green), and estimate how many of the 300 marbles are red. Answer: P (red) = \frac{54}{120} = \frac{9}{20} = 0.45 P (blue or green) = \frac{42 + 24}{120} = \frac{66}{120} = \frac{11}{20} = 0.55 P (not green) = 1 - \frac{24}{120} = \frac{96}{120} = \frac{4}{5} = 0.8 Red marbles \approx 0.45 \times 300 = 135

Your turn — try these on paper, then check the answers below. 1 . A letter is chosen from LETTER . Find P (T) . 2 . Find the probability of drawing a heart from a pack of 52 cards. 3 . A coin was tossed 80 times and landed heads 44 times. Estimate P (heads) and P (tails) .

Example 1 — Answers to Your Turn

1 . T appears twice in 6 letters: \frac{2}{6} = \frac{1}{3} 2 . \frac{13}{52} = \frac{1}{4} 3 . P (heads) = \frac{44}{80} = 0.55; P (tails) = 0.45

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