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Q5:
A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are there?
Solution :
Step 1: Analyzing the Sample Space for a Single Event
Each individual toss of a coin is an independent experiment. A fair coin has two possible outcomes for each toss, defined as the set $S = \{H, T\}$, where $H$ denotes 'Heads' and $T$ denotes 'Tails'.
Thus, the number of possible outcomes for a single toss, $n_1 = 2$.
Step 2: Applying the Fundamental Counting Principle
The Fundamental Counting Principle (also known as the Multiplication Principle) states that if there are $n$ ways to perform one task and $m$ ways to perform a second task, the total number of ways to perform both is $n \times m$. [Per the Fundamental Counting Principle of Combinatorics].
For $k$ independent events, the total number of outcomes is given by the product of the number of outcomes for each event: $N = n_1 \times n_2 \times n_3 \times \dots \times n_k$.
Step 3: Calculating Total Outcomes for Three Tosses
Given the coin is tossed $k = 3$ times:
- Number of outcomes for toss 1 ($n_1$) = 2
- Number of outcomes for toss 2 ($n_2$) = 2
- Number of outcomes for toss 3 ($n_3$) = 2
Total possible outcomes = $2 \times 2 \times 2 = 2^3$.
Total possible outcomes = $8$.
Visual Representation of the Sample Space
Step 4: Enumerating the Sample Space ($S$)
To verify the calculation, we list the Cartesian product of the three sets $\{H, T\} \times \{H, T\} \times \{H, T\}$:
| Outcome Index | Sample Point |
|---|---|
| 1 | (H, H, H) |
| 2 | (H, H, T) |
| 3 | (H, T, H) |
| 4 | (H, T, T) |
| 5 | (T, H, H) |
| 6 | (T, H, T) |
| 7 | (T, T, H) |
| 8 | (T, T, T) |
Final Solution: The total number of possible outcomes when a coin is tossed 3 times is $2^3 = 8$.
More Questions from Class 11 Mathematics Permutations and Combinations EXERCISE 6.1
- Q1(i): How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that (i) repetition of the digits is allowed?
- Q1(ii): How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that (ii) repetition of the digits is not allowed?
- Q2: How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?
- Q3: How many 4-letter code can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?
- Q4: How many 5-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?
- Q6: Given 5 flags of different colours, how many different signals can be generated if each signal requires the use of 2 flags, one below the other?
CBSE Solutions for Class 11 Mathematics Permutations and Combinations
Chapters in CBSE - Class 11 Mathematics
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Other Subjects in CBSE - Class 11
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