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🎲 Chapter 7: The Mathematics of Maybe: Introduction to Probability • Exercise Set 7.2

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Exercise Set 7.1 1Exercise Set 7.2 4Exercise Set 7.3 3Exercise Set 7.4 1End-of-Chapter Exercises 8All Exercises 17
Q1

A teacher mixes a large bag of sweets of different colours and randomly selects a sample of 30 sweets. She counts: 10 red | 8 green | 7 yellow | 5 blue

  • (i) Calculate the probability that a randomly picked sweet from the sample is green.
  • (ii) If there are 600 sweets in total in the large bag, estimate how many are likely to be yellow, based on the sample results.
  • (i) Probability of green = Number of green / Total in sample = $8 / 30 = \mathbf{4/15}$ (or $\approx 0.267$).
  • (ii) Experimental probability of yellow = $7 / 30$.
    Estimated yellow sweets in total bag = $(7 / 30) \times 600 = 7 \times 20 = \mathbf{140}$ sweets.
Q2

A survey is conducted at a school where a random sample of 40 students is asked about their favourite club. The responses are: 14 Science | 11 Arts | 9 Sports | 6 Debate. Assume there are 800 students in the whole school.

  • (i) What is the probability that a randomly chosen student from the sample prefers the Arts Club?
  • (ii) Using the sample results, estimate how many students in the whole school are likely to prefer the Sports Club.
  • (i) Probability of Arts Club = $11 / 40$ (or $0.275$).
  • (ii) Experimental probability of Sports Club = $9 / 40$.
    Estimated students preferring Sports Club = $(9 / 40) \times 800 = 9 \times 20 = \mathbf{180}$ students.
Q5

What is the probability of getting an even number when rolling a fair 6-sided die?

Answer: The possible outcomes are $\{1, 2, 3, 4, 5, 6\}$.
The even numbers are $\{2, 4, 6\}$.
Probability = Number of favourable outcomes / Total possible outcomes = $3 / 6 = \mathbf{1/2}$ (or $0.5$).

Q6

Suppose you roll a 6-sided die 12 times and get a '3' three times.

  • (i) What is the experimental probability of rolling a '3'?
  • (ii) What is the theoretical probability of rolling a '3'?
  • (iii) Why might these probabilities be different? What would you expect to happen if you roll the die 60, 600, or 6000 times?
  • (i) Experimental probability = $3 / 12 = \mathbf{1/4}$ (or $0.25$).
  • (ii) Theoretical probability = $\mathbf{1/6}$ (or $\approx 0.167$).
  • (iii) They are different because experimental probability is based on a small number of actual trials, which are subject to randomness and short-term fluctuations. If you roll the die 60, 600, or 6000 times, the Law of Large Numbers states that the experimental probability will gradually get closer and closer to the theoretical probability of $1/6$.