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. To find the total number of possible outcomes, we multiply the number of outcomes for each toss: 2 outcomes × 2 outcomes × 2 outcomes = $2^3$ = 8 outcomes.

The sample space (S) is the set of all these possible outcomes. Therefore, the sample space is:

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

When a standard six-sided die is thrown once, the possible outcomes are the numbers 1, 2, 3, 4, 5, or 6.

When the die is thrown two times, the outcome of each throw is recorded. We can represent the sample space as a set of ordered pairs $(x,\; y)$, where $x$ is the result of the first throw and $y$ is the result of the second throw. Both $x$ and $y$ can be any integer from 1 to 6.

The total number of possible outcomes is the product of the number of outcomes for each throw: 6 outcomes × 6 outcomes = $6\; \backslash times\; 6\; =\; 36$.

The sample space $S$ is given by:

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

Listing all 36 pairs explicitly:

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

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 calculated by multiplying the number of outcomes for each toss: 2 × 2 × 2 × 2 = $2^4$ = 16 outcomes.

The sample space (S) is the collection of all these distinct possible outcomes. Thus, the sample space is:

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

A coin has two possible outcomes: Head (H) and Tail (T).

A standard six-sided die has six possible outcomes: the numbers 1, 2, 3, 4, 5, or 6.

When a coin is tossed and a die is thrown simultaneously, each outcome consists of a result from the coin toss paired with a result from the die throw. The total number of possible outcomes is the product of the number of outcomes for each event: 2 outcomes (coin) × 6 outcomes (die) = 12 outcomes.

The sample space (S) is the set of all these combined outcomes:

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