Basic Concepts of Probability and Sample Space

Probability measures uncertainty numerically. It helps us answer questions such as:

• What is the chance of getting heads when a coin is tossed?

• What is the chance of rolling a 6 on a fair die?

• What is the chance of drawing an ace from a pack of cards?

Important: Probability does not tell us what will happen; it tells us how likely an event is to happen.

Random Experiment: An experiment in which all possible results are known in advance, but the result of any particular performance cannot be predicted with certainty.

Key Features

• All possible outcomes are known before the experiment.

• The exact outcome of one trial cannot be predicted.

• The experiment can usually be repeated under similar conditions.

Outcome: The result of a random experiment

Important: An outcome is a single possible result, not the complete list of results.

Equally Likely Outcomes: Outcomes are equally likely when there is no reason to expect one outcome more often than another.

Conditions

• The coin, die, cards, or selection method should be fair or unbiased.

• Each outcome should have the same chance of occurring.

Sample Space: The set of all possible outcomes of a random experiment. It is usually denoted by S.

Sample Point: Each element of the sample space is called a sample point.

Number of Sample Points: If S is a sample space, then n(S) denotes the number of sample points in S.

\[ \begin{array}{|c|c|c|c|} \hline \text{S.No.} & \text{Random experiment} & \text{Sample space} & \text{Number of sample points in } S \\ \hline 1 & \text{One coin is tossed} & S = \{H, T\} & n(S) = 2 \\ \hline 2 & \text{Two coins are tossed} & S = \{HH, HT, TH, TT\} & n(S) = 4 \\ \hline 3 & \text{Three coins are tossed} & \begin{array}{c} S = \{HHH, HHT, HTH, THH, HTT, THT, \\ TTH, TTT\} \end{array} & n(S) = 8 \\ \hline 4 & \text{A die is thrown} & S = \{1, 2, 3, 4, 5, 6\} & n(S) = 6 \\ \hline 5 & \text{Two dice are thrown} & \begin{array}{c} S = \{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), \\ (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), \\ (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), \\ (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), \\ (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), \\ (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} \end{array} & n(S) = 36 \\ \hline 6 & \begin{array}{c} \text{A card is drawn from a pack} \\ \text{bearing numbers from 1 to 25} \end{array} & S = \{1, 2, 3, 4, \cdots\cdots\cdots\cdots, 25\} & n(S) = 25 \\ \hline 7 & \begin{array}{c} \text{A card is drawn from a well} \\ \text{shuffled pack of 52 playing} \\ \text{cards.} \end{array} & \begin{array}{l} \text{Diamond : Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack,} \\ \text{Queen, King} \\ \text{Spade : Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack,} \\ \text{Queen, King} \\ \text{Heart : Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack,} \\ \text{Queen, King} \\ \text{Club : Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack,} \\ \text{Queen, King} \end{array} & n(S) = 52 \\ \hline \end{array} \]

• A random experiment has known possible outcomes but an unpredictable result.

• An outcome is one possible result of an experiment.

• Outcomes are equally likely when each has the same chance.

• The sample space S is the set of all possible outcomes.

• A sample point is one element of the sample space.

• n(S) is the number of outcomes in the sample space.

• One coin: n(S) = 2-; two coins: n(S) = 4; three coins: n(S) = 8.

• One die: n(S) = 6; two dice: n(S) = 36.

• A standard card pack has 52 cards, 4 suits, 12 face cards, and 4 aces.