Mathematics • Year 7 • Unit 4 • Lesson 17
Theoretical Probability — Real World
Apply P = f/n and the complement rule to real situations: a class raffle, a card game, a school carnival spinner, a multiple-choice quiz and a lottery scratch card.
1. Word problems
Show sample space and formula line before stating answers.
1.1 — Class raffle. Mrs Patel runs a raffle for the 30 students in 7B. Each student writes their name on a card and one card is drawn at random. There are 4 students named "Lily" and 6 students named "Hassan" (Lily and Hassan are common names in 7B). (a) Find P(drawing a Lily). (b) Find P(drawing a Hassan or a Lily). (c) Find P(drawing neither Lily nor Hassan) — use the complement rule. 4 marks
1.2 — Card game. A friend deals one card from a well-shuffled 52-card deck. (a) Find P(red card). (b) Find P(picture card — Jack, Queen or King, any suit). (c) Find P(not a picture card) using the complement rule. 4 marks
1.3 — School carnival spinner. A spinner is divided into 8 equal sectors labelled with the prizes: Drink, Drink, Drink, Sticker, Sticker, Pen, Cap, JACKPOT. One spin per ticket. (a) Find P(JACKPOT). (b) Find P(Drink). (c) Find P(any prize that is NOT a sticker) using the complement rule. 4 marks
1.4 — Multiple-choice guessing. A multiple-choice question has 4 options (A, B, C, D), only one is correct. A student guesses at random. (a) Find P(correct). (b) Find P(wrong) using the complement rule. (c) On a 20-question test, if every answer is guessed, what is the expected number of correct answers? 3 marks
1.5 — Lottery scratch card. A 'Pick A Letter' scratch card has 26 covered squares — one for each letter of the alphabet. Hidden under the letters A, E and Y is a prize. You scratch one square. (a) Find P(win). (b) Find P(not winning). (c) Explain in one sentence why theoretical probability can be calculated here without doing any experiment. 4 marks
2. Explain your thinking
Communication matters. Use full sentences. 4 marks
2.1 A Year 7 student says: "There are only two outcomes when you buy a lottery ticket — you win, or you don't. So P(win) = 1/2." In your own words, explain (i) why the student's reasoning breaks the equally-likely rule, (ii) why we must use experimental probability (or the lottery's own published odds) instead of P = f/n, and (iii) what the right way is to think about whether outcomes are equally likely.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Class raffle
(a) P(Lily) = 4/30 = 2/15 ≈ 0.133.
(b) P(Hassan or Lily) = (6 + 4)/30 = 10/30 = 1/3 ≈ 0.333.
(c) P(neither) = 1 − 1/3 = 2/3 ≈ 0.667.
1.2 — Card game
(a) P(red) = 26/52 = 1/2 = 0.5.
(b) P(picture) = 12/52 = 3/13 ≈ 0.231.
(c) P(not picture) = 1 − 3/13 = 10/13 ≈ 0.769.
1.3 — Carnival spinner (8 equal sectors)
(a) P(JACKPOT) = 1/8 = 0.125.
(b) P(Drink) = 3/8 = 0.375.
(c) P(sticker) = 2/8 = 1/4. P(not sticker) = 1 − 1/4 = 3/4 = 0.75.
1.4 — Multiple-choice guessing
(a) P(correct) = 1/4 = 0.25.
(b) P(wrong) = 1 − 1/4 = 3/4 = 0.75.
(c) Expected correct = 0.25 × 20 = 5 correct.
1.5 — Scratch card
(a) P(win) = 3/26 ≈ 0.115.
(b) P(not winning) = 1 − 3/26 = 23/26 ≈ 0.885.
(c) The 26 squares are equally likely to be chosen and we know the exact number of winners (3) and the exact total (26), so we can apply P = f/n directly — no experiment required.
2.1 — Explain your thinking (sample response)
(i) The student has assumed that "two outcomes" automatically means each is 1/2. The formula P = f/n only works if all outcomes are equally likely, and "winning the lottery" is nowhere near as likely as "not winning".
(ii) For lotteries, the true probability is calculated from how many winning tickets exist out of millions of possible tickets — that ratio is published, not 1/2. If we did not have those numbers, we would estimate probability experimentally from a large number of trials, not by assuming.
(iii) The right check is: are the basic outcomes (each individual ticket, each face of a die, each marble in a bag) physically equally likely? If yes, group them into events and use P = f/n. If no, we must use experimental probability or quoted odds instead.
Marking: 1 for naming the equally-likely error, 1 for naming the right method for lotteries, 1 for clear sentences, 1 for linking back to lesson rules.