The Mathematics of Maybe: Introduction to Probability | Exercise 7.3

Question 2

For the following experiments write down the sample space SS.

(i) Rolling a die and tossing a coin together.

(ii) Choosing a random integer between −5-5 and +5+5.

(iii) A box containing 5 green and 7 red balls. One ball is drawn at random.

Check your answer with HomiSolve it yourself, then let Homi check your steps and spot mistakes.
Solution
Understand the Question
  • The sample space (SS) of a random experiment is the set of all possible outcomes.
  • To write the sample space:
    • Identify all possible individual outcomes for each component of the experiment.
    • Combine or list all distinct outcomes as a set enclosed in curly braces {… }\{\dots\}.

(i) Rolling a die and tossing a coin together.

Step 1 · List all combined outcomes

Outcomes for rolling a die: {1,2,3,4,5,6}\{1, 2, 3, 4, 5, 6\}

Outcomes for tossing a coin: {H,T}\{H, T\}

Combining each die outcome with each coin outcome gives 6×2=126 \times 2 = 12 possible pairs: S={(1,H),(1,T),(2,H),(2,T),(3,H),(3,T),(4,H),(4,T),(5,H),(5,T),(6,H),(6,T)}S = \{(1, H), (1, T), (2, H), (2, T), (3, H), (3, T), (4, H), (4, T), (5, H), (5, T), (6, H), (6, T)\}

Answer

(i) S={(1,H),(1,T),(2,H),(2,T),(3,H),(3,T),(4,H),(4,T),(5,H),(5,T),(6,H),(6,T)}S = \{(1, H), (1, T), (2, H), (2, T), (3, H), (3, T), (4, H), (4, T), (5, H), (5, T), (6, H), (6, T)\}

(ii) Choosing a random integer between −5-5 and +5+5.

Step 1 · List integers in the given range

Listing all integers from −5-5 to +5+5 (inclusive): S={−5,−4,−3,−2,−1,0,1,2,3,4,5}S = \{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}

Total number of outcomes =11= 11.

Answer

(ii) S={−5,−4,−3,−2,−1,0,1,2,3,4,5}S = \{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}

(iii) A box containing 5 green and 7 red balls. One ball is drawn at random.

Step 1 · Identify the possible color outcomes

The box contains two types of balls: green (GG) and red (RR). When a single ball is drawn, the outcome in terms of color is: S={G,R}S = \{G, R\}

(Note: If individual balls are treated as distinct/distinguishable, the sample space can also be written as S={G1,G2,G3,G4,G5,R1,R2,R3,R4,R5,R6,R7}S = \{G_1, G_2, G_3, G_4, G_5, R_1, R_2, R_3, R_4, R_5, R_6, R_7\}.)

Answer

(iii) S={G,R}S = \{G, R\}

Common Mistakes
  • Omitting 00: Forgetting to include 00 when listing integers between negative and positive limits.
  • Missing Pairs in Product Spaces: Missing some combinations in compound experiments; verify the total count using the multiplication principle (6×2=126 \times 2 = 12).
  • Distinguishable vs. Indistinguishable Outcomes: In ball-drawing problems, clearly note whether the sample space refers to the distinct colors {G,R}\{G, R\} or individual numbered balls {G1,…,G5,R1,…,R7}\{G_1, \dots, G_5, R_1, \dots, R_7\}.

More questions in Exercise 7.3

Q1

When a single 6-sided die is rolled, what is the total number of possible outcomes in the sample space?

Q2

For the following experiments write down the sample space SS.

(i) Rolling a die and tossing a coin together.

(ii) Choosing a random integer between −5-5 and +5+5.

(iii) A box containing 5 green and 7 red balls. One ball is drawn at random.

Q3

In a village fair, there are 3 popular snacks available: Samosa, Pakora, and Bhaji. For drinks, villagers can choose either Chai or Lassi.

(i) List the sample space of all possible snack and drink combinations a person could choose at the fair.

(ii) List the event 'Selecting Samosa as a snack.'

← Back to The Mathematics of Maybe: Introduction to Probability