Exercise 7.3 Practice
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Overview
This page provides comprehensive Ch 7: The Mathematics of Maybe: Introduction to Probability - Exercise Set 7.3 Practice. Practice writing complete sample spaces $S$ and event subsets for combinations of coin-die tosses, integer intervals, colored balls, and fair snacks/drinks with step-by-step solutions.
Sample Spaces, Event Subsets & Multidisciplinary Outcomes
Q1: Single Die Outcomes size
When a single 6-sided die is rolled, what is the total number of possible outcomes in the sample space?
The outcomes when rolling a standard 6-sided die are $\{1, 2, 3, 4, 5, 6\}$.
Therefore, the total number of possible outcomes in the sample space $n(S)$ is **$6$**.
Therefore, the total number of possible outcomes in the sample space $n(S)$ is **$6$**.
Total outcomes = 6
Q2: Write Sample Spaces
For the following experiments write down the sample space $S$.
(i) Rolling a die and tossing a coin together.
(ii) Choosing a random integer between $-5$ and $+5$.
(iii) A box containing 5 green and 7 red balls. One ball is drawn at random.
(i) Rolling a die and tossing a coin together.
(ii) Choosing a random integer between $-5$ and $+5$.
(iii) A box containing 5 green and 7 red balls. One ball is drawn at random.
(i) Rolling a die and tossing a coin together:
Let die outcomes be $\{1, 2, 3, 4, 5, 6\}$ and coin outcomes be $\{H, T\}$.
$$S = \mathbf{\{(1,H), (1,T), (2,H), (2,T), (3,H), (3,T), (4,H), (4,T), (5,H), (5,T), (6,H), (6,T)\}}$$
Size $n(S) = 6 \times 2 = 12$.
Let die outcomes be $\{1, 2, 3, 4, 5, 6\}$ and coin outcomes be $\{H, T\}$.
$$S = \mathbf{\{(1,H), (1,T), (2,H), (2,T), (3,H), (3,T), (4,H), (4,T), (5,H), (5,T), (6,H), (6,T)\}}$$
Size $n(S) = 6 \times 2 = 12$.
(ii) Choosing a random integer between $-5$ and $+5$:
Assuming inclusive boundaries:
$$S = \mathbf{\{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}}$$
Size $n(S) = 11$. (If exclusive, $S = \{-4, -3, -2, -1, 0, 1, 2, 3, 4\}$, size $n(S) = 9$).
Assuming inclusive boundaries:
$$S = \mathbf{\{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}}$$
Size $n(S) = 11$. (If exclusive, $S = \{-4, -3, -2, -1, 0, 1, 2, 3, 4\}$, size $n(S) = 9$).
(iii) Box with 5 green and 7 red balls:
If we only track the color of the drawn ball:
$$S = \mathbf{\{Green, Red\}}$$
If we distinguish the individual balls (with subscripts):
$$S = \mathbf{\{G_1, G_2, G_3, G_4, G_5, R_1, R_2, R_3, R_4, R_5, R_6, R_7\}}$$
If we only track the color of the drawn ball:
$$S = \mathbf{\{Green, Red\}}$$
If we distinguish the individual balls (with subscripts):
$$S = \mathbf{\{G_1, G_2, G_3, G_4, G_5, R_1, R_2, R_3, R_4, R_5, R_6, R_7\}}$$
(i) 12 pairs {(1,H)...(6,T)}
(ii) \{-5...5\} (size 11)
(iii) \{Green, Red\}
Q3: Fair Snack and Drink combinations
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’.
(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’.
(i) Sample Space $S$:
Let snacks be $\{S, P, B\}$ and drinks be $\{C, L\}$.
$$S = \mathbf{\{(Samosa, Chai), (Samosa, Lassi), (Pakora, Chai), (Pakora, Lassi), (Bhaji, Chai), (Bhaji, Lassi)\}}$$
Size $n(S) = 3 \times 2 = 6$.
Let snacks be $\{S, P, B\}$ and drinks be $\{C, L\}$.
$$S = \mathbf{\{(Samosa, Chai), (Samosa, Lassi), (Pakora, Chai), (Pakora, Lassi), (Bhaji, Chai), (Bhaji, Lassi)\}}$$
Size $n(S) = 3 \times 2 = 6$.
(ii) Event $E$ ‘Selecting Samosa as a snack’:
Filter the outcomes of $S$ where the snack component is Samosa:
$$E = \mathbf{\{(Samosa, Chai), (Samosa, Lassi)\}}$$
Size $n(E) = 2$.
Filter the outcomes of $S$ where the snack component is Samosa:
$$E = \mathbf{\{(Samosa, Chai), (Samosa, Lassi)\}}$$
Size $n(E) = 2$.
(i) 6 combinations (ii) Event E = {(Samosa, Chai), (Samosa, Lassi)}