Experiment, Sample space and Events - Basics

Definition - (Experiment) An experiment is the process whose outcome is not predictable with certainty in advance.

Example

1. Tossing a coin once or several times
2. Tossing a six-faced die
3. Selecting a card or cards from a deck
4. Obtaining blood types from a group of individuals
5. Measuring (in hours) the lifetime of a transistor

Definition - (Sample Space) The set of all possible outcomes of an experiment is called the experiment's sample space and is denoted by S.

Example

1. If the experiment is tossing a coin, then the outcomes will be head and tail, and hence the sample space is S = {H, T}.
2. If the experiment consists of flipping two coins, then the outcomes will be heads, head and tail and both tails. So the sample space S consists of the following four points: S = {(H, H),(H, T),(T, H),(T, T)}.
3. If the experiment consists of tossing a die, then the sample space is S = {1, 2, 3, 4, 5, 6}.
4. If the experiment consists of tossing two dice, then the sample space consists of the 36 points S = {(i, j): i, j = 1, 2, 3, 4, 5, 6}, where the outcome (2, 5) means two appears in the first die and 5 on the other die.
5. If the experiment consists of measuring (in hours) the transistor's lifetime, then the sample space consists of all nonnegative real numbers. That is, S = {x | 0 ≤ x < ∞}

Definition - (Event) Any subset E of the sample space S is known as an event. An event is called elementary or straightforward if it consists of a single outcome. If the experiment's outcome is contained in E, then we say that the event E has occurred.

Example

1. In example 2, E = {(H, H),(H, T)} is the event that a head appears on the first coin. 2. In example 4, E1 = {(1, 5),(2, 4),(3, 3),(2, 4),(1, 5)} is the event that the sum of the dice equals 6. The Event E2 that the sum of dice equals 12 is the simple event as E2 = {(6, 6)} consists of only one outcome.
2. In example 5 , E = {x | x ≥ 10} is the event that the transistor lasts at least 10 hours.

For any two events A and B of the sample space S, we can define the new events

• A ∪ B = {ω ∈ S | ω ∈ A or ω ∈ B} consists of all outcomes that are either in A or in B or A and B.

• A ∩ B = {ω ∈ S | ω ∈ A and ω ∈ B} consists of all outcomes that are both in A and B

• More generally, S∞ i=1 Ai = {ω ∈ S | ω ∈ Ai for some i} consists of all outcomes which lie in at least one event Ai and T∞ i=1 Ai = {ω ∈ S | ω ∈ Ai for all i} consists of all outcomes which lie in all events Ai

• Ac = {ω ∈ S | ω /∈ A} consists of all outcomes, not in A.

• An event that doesn't contain any outcome is referred to as a null event and is denoted by ∅.

• Sample space is also an event referred to as a specific or sure event.

Definition: (Mutually Exclusive) Two events A and B of the sample space S are said to be mutually exclusive if A ∩ B = ∅. i.e. if event A occurs, then event B doesn't occur, and vice versa.

Example: If S is the sample space of tossing a die, then the events A = {ω ∈ S | ω is even} and B = {ω ∈ S | ω is odd} are mutually exclusive.

Definition: Let A and B be two events of the sample space S. If A ⊂ B, then the occurrence of A necessarily implies the occurrence of B, i.e., the event B occurs whenever the event A occurs.

Types of Events:

1. Elementary event or simple event
2. Impossible event
3. Sure or a specific event
4. Mutually Exclusive Events
5. The complement of an event
6. Exhaustive events