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🎲 Chapter 7: The Mathematics of Maybe: Introduction to Probability • End-of-Chapter Exercises

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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

Fill in the blanks.

  • (i) The probability of an impossible event is ________________.
  • (ii) The set of all possible outcomes of a random experiment is called the ________________.
  • (iii) The probability of an event that is certain to happen is ________________.
  • (iv) Tossing a fair coin has a probability of ________________ for getting heads.
  • (i) 0
  • (ii) sample space
  • (iii) 1
  • (iv) 1/2 (or 0.5)
Q4

Write the sample space and calculate the probability based on the given information.

  • (i) Two coins are tossed at the same time. What is the probability of getting at least one head?
  • (ii) Ten identical cards numbered 1 to 10 are placed in a box. One card is drawn at random. What is the probability of drawing a card with an even number?
  • (iii) A die is rolled once. What is the probability of getting a number greater than 4?
  • (iv) A bag contains 3 red balls, 2 blue balls, and 1 green ball. One ball is picked at random. What is the probability that it is not red?
  • (v) Three coins are tossed simultaneously. What is the probability of getting exactly two heads?
  • (i) $S = \{HH, HT, TH, TT\}$. Favourable = $\{HH, HT, TH\}$. Probability = $\mathbf{3/4}$.
  • (ii) $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. Favourable (even) = $\{2, 4, 6, 8, 10\}$. Probability = $5/10 = \mathbf{1/2}$.
  • (iii) $S = \{1, 2, 3, 4, 5, 6\}$. Favourable (>4) = $\{5, 6\}$. Probability = $2/6 = \mathbf{1/3}$.
  • (iv) $S = \{\text{Red, Red, Red, Blue, Blue, Green}\}$. Favourable (not red) = 3 (2 blue + 1 green). Probability = $\mathbf{3/6 = 1/2}$.
  • (v) $S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$. Favourable (exactly two heads) = $\{HHT, HTH, THH\}$. Probability = $\mathbf{3/8}$.
Q6

A child has 2 shirts (one red and one blue) and 3 types of pants (jeans, khakis, and shorts). List all the possible combinations of outfits consisting of one shirt and one pair of pants. Display your answer in a table format.

ShirtPantsOutfit Combination
RedJeansRed & Jeans
RedKhakisRed & Khakis
RedShortsRed & Shorts
BlueJeansBlue & Jeans
BlueKhakisBlue & Khakis
BlueShortsBlue & Shorts
Q7

A tyre company records distances before replacement in 1000 cases.

Distance (km)Less than 40004001 to 90009001 to 14000More than 14000
Number of cases20210325445

Find the probability that a randomly chosen tyre lasts:
(i) Less than 4000 km.
(ii) Between 4000 and 14000 km.
(iii) More than 14000 km.

  • (i) Probability = $20 / 1000 = \mathbf{0.02}$ (or 2%).
  • (ii) Favourable cases = $210 + 325 = 535$. Probability = $535 / 1000 = \mathbf{0.535}$ (or 53.5%).
  • (iii) Probability = $445 / 1000 = \mathbf{0.445}$ (or 44.5%).
Q8

The letters of the word 'PEACE' are placed on cards. Leela draws a card without looking.

  • (i) What is the probability that it is a P, E or C?
  • (ii) What is the probability that it is not an E?
  • (i) Total letters = 5. Favourable (P, E, or C) = 4 (P, E, E, C). Probability = $\mathbf{4/5}$.
  • (ii) Favourable (not E) = 3 (P, A, C). Probability = $\mathbf{3/5}$.
Q9

A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8, and these are equally likely outcomes. What is the probability that it will point at

  • (i) 8?
  • (ii) An odd number?
  • (iii) A number greater than 2?
  • (iv) A number less than 9?
  • (v) A multiple of 3?
1 2 3 4 5 6 7 8
8-Sector Equally Likely Spinner Wheel
  • (i) Favourable = 1. Probability = $\mathbf{1/8}$.
  • (ii) Odd numbers = $\{1, 3, 5, 7\}$ (4 outcomes). Probability = $4/8 = \mathbf{1/2}$.
  • (iii) Numbers > 2 = $\{3, 4, 5, 6, 7, 8\}$ (6 outcomes). Probability = $6/8 = \mathbf{3/4}$.
  • (iv) Numbers < 9 = All 8 numbers. Probability = $8/8 = \mathbf{1}$ (Certain).
  • (v) Multiples of 3 = $\{3, 6\}$ (2 outcomes). Probability = $2/8 = \mathbf{1/4}$.
Q10

A basket contains 4 red balls and 5 blue balls. One ball is drawn and laid aside, and a second ball is drawn. Draw a tree diagram to represent the possible outcomes and probabilities. Use the tree diagram to answer the following questions.

  • (i) What is the probability of drawing a red ball and then a blue ball?
  • (ii) What is the probability of drawing 2 blue balls?

Answer:
Tree Diagram Construction:
First draw: P(Red) = 4/9, P(Blue) = 5/9.
Second draw (without replacement, total is now 8 balls):
- If Red was drawn first: P(Red) = 3/8, P(Blue) = 5/8.
- If Blue was drawn first: P(Red) = 4/8, P(Blue) = 4/8.

(i) P(Red then Blue) = P(Red first) $\times$ P(Blue second | Red first) = $(4/9) \times (5/8) = 20/72 = \mathbf{5/18}$.
(ii) P(Blue then Blue) = P(Blue first) $\times$ P(Blue second | Blue first) = $(5/9) \times (4/8) = 20/72 = \mathbf{5/18}$.

1st Draw (9 balls) 2nd Draw (8 balls) Outcome & Prob 4/9 5/9 Red₁ Blue₁ 3/8 5/8 Red₂ Blue₂ 4/8 4/8 Red₂ Blue₂ RR: 12/72 = 1/6 RB: 20/72 = 5/18 BR: 20/72 = 5/18 BB: 20/72 = 5/18
Tree Diagram for two draws without replacement (4 Red, 5 Blue balls)
Q15

Three coins are tossed, and the number of heads is recorded. Which of the following lists is a sample space for this experiment? Why do the other lists fail to qualify as a sample space?

  • (i) {1, 2, 3}
  • (ii) {0, 1, 2}
  • (iii) {0, 1, 2, 3, 4}
  • (iv) {0, 1, 2, 3}

Answer: The correct list is (iv) {0, 1, 2, 3}. When tossing three coins, you can get 0 heads (TTT), 1 head (HTT, THT, TTH), 2 heads (HHT, HTH, THH), or 3 heads (HHH).
The other lists fail because:
(i) It omits the possibility of getting 0 heads.
(ii) It omits the possibility of getting 3 heads.
(iii) It includes 4 heads, which is impossible when only tossing three coins.