CBSE Class 11 Mathematics Chapter 16: Probability NCERT Solutions

A coin has two possible outcomes when tossed: a Head (H) or a Tail (T).

When a coin is tossed three times, each toss is an independent event. The total number of possible outcomes is calculated by multiplying the number of outcomes for each toss: $2 \times 2 \times 2 = 2^3 = 8$.

The sample space (S) is the set of all these possible outcomes. It can be listed systematically as follows:

$$S\; =\; \backslash \{HHH,\; HHT,\; HTH,\; HTT,\; THH,\; THT,\; TTH,\; TTT\backslash \}$$

When a standard six-sided die is thrown, the possible outcomes are the integers from 1 to 6, i.e., {1, 2, 3, 4, 5, 6}.

When a die is thrown two times, the outcome of the first throw and the outcome of the second throw are recorded as an ordered pair $(x, y)$. Here, $x$ represents the outcome of the first throw and $y$ represents the outcome of the second throw.

Both $x$ and $y$ can take any value from 1 to 6. Therefore, the sample space $S$ is the set of all possible ordered pairs:

$$S\; =\; \backslash \{(x,\; y):\; x,\; y\; =\; 1,\; 2,\; 3,\; 4,\; 5,\; 6\backslash \}$$

The total number of elements (outcomes) in this sample space is the product of the number of possibilities for each throw: $6 \times 6 = 36$.

The sample space is explicitly listed as:

$$S\; =\; \backslash \{(1,\; 1),\; (1,\; 2),\; (1,\; 3),\; (1,\; 4),\; (1,\; 5),\; (1,\; 6),\; \backslash \backslash \; (2,\; 1),\; (2,\; 2),\; (2,\; 3),\; (2,\; 4),\; (2,\; 5),\; (2,\; 6),\; \backslash \backslash \; (3,\; 1),\; (3,\; 2),\; (3,\; 3),\; (3,\; 4),\; (3,\; 5),\; (3,\; 6),\; \backslash \backslash \; (4,\; 1),\; (4,\; 2),\; (4,\; 3),\; (4,\; 4),\; (4,\; 5),\; (4,\; 6),\; \backslash \backslash \; (5,\; 1),\; (5,\; 2),\; (5,\; 3),\; (5,\; 4),\; (5,\; 5),\; (5,\; 6),\; \backslash \backslash \; (6,\; 1),\; (6,\; 2),\; (6,\; 3),\; (6,\; 4),\; (6,\; 5),\; (6,\; 6)\backslash \}$$

A coin has two possible outcomes for each toss: Head (H) or Tail (T).

When a coin is tossed four times, the total number of possible outcomes is $2 \times 2 \times 2 \times 2 = 2^4 = 16$.

The sample space (S) is the set containing all these 16 distinct outcomes. Listing them systematically gives:

$$S\; =\; \backslash \{HHHH,\; HHHT,\; HHTH,\; HHTT,\; HTHH,\; HTHT,\; HTTH,\; HTTT,\; THHH,\; THHT,\; THTH,\; THTT,\; TTHH,\; TTHT,\; TTTH,\; TTTT\backslash \}$$

The experiment involves two independent actions: tossing a coin and throwing a die.

The possible outcomes for tossing a coin are Head (H) and Tail (T).

The possible outcomes for throwing a standard six-sided die are the numbers {1, 2, 3, 4, 5, 6}.

To find the sample space for this combined experiment, we pair each outcome of the coin toss with each outcome of the die throw. The sample space (S) is the set of all such pairs:

$$S\; =\; \backslash \{H1,\; H2,\; H3,\; H4,\; H5,\; H6,\; T1,\; T2,\; T3,\; T4,\; T5,\; T6\backslash \}$$

There are $2 \times 6 = 12$ possible outcomes in this sample space.