Basic Probability and event types

Random Experiment: The event whichs outcome is not predefinite.

Sample Space(Set Representation): Set of all possible outcome. Denoted by S.
Sample Space = {All possible outcome}

Probability : Chance of occuring.

Practice Quesiton: A coin is tossed. If the result is head, a die is thrown. If die shows up and even number, the die is thrown again. What is the sample space for this experimenet.

Solution:

Let's List of All Possible Outcomes:

1. If the coin shows T:
   • Outcome: T
2. If the coin shows H and the first die roll is 1, 3, or 5:
   • Outcomes: H1, H3, H5
3. If the coin shows H and the first die roll is 2:
   • Second die roll can be 1, 2, 3, 4, 5, or 6
   • Outcomes: H21, H22, H23, H24, H25, H26
4. If the coin shows H and the first die roll is 4:
   • Second die roll can be 1, 2, 3, 4, 5, or 6
   • Outcomes: H41, H42, H43, H44, H45, H46
5. If the coin shows H and the first die roll is 6:
   • Second die roll can be 1, 2, 3, 4, 5, or 6
   • Outcomes: H61, H62, H63, H64, H65, H66

So, Sample Space = {T, H1, H3, H5, H21, H22, H23, H24, H25, H26, H41, H42, H43, H44, H45, H46, H61, H62, H63, H64, H65, H66}

Venn diagram in probability

Venn Diagrams: A Digram where sample space is prepresented by rectangle and the elements of the sample space by points. Subsets or events of the sample space are represented by the region(circle) withing the rectangle.

Event and Its types

Event: Every subset of sample space.

The probability of an event E occuring is given by:
P(E) = $\frac{Number of favourable outcomes}{Total number of possible outcomes}$

Types of Events in Probability:

1. Simple Event: A simple event consists of a single outcome.
   Example:- Rolling a die and getting a 3.
   Sample Space: $S = \{1, 2, 3, 4, 5, 6\}$
   Simple Event: $E = \{3\}$
2. Compound Event: A compound event consists of two or more simple events.
   Example:- Rolling a die and getting an even number.
   Sample Space: $S = \{1, 2, 3, 4, 5, 6\}$
   Compound Event: $E = \{2, 4, 6\}$
3. Mutually Exclusive Events: Two events are mutually exclusive if they cannot occur at the same time.
   Example:- Drawing a card from a deck and getting either a King or a Queen.
   Sample Space: S = all 52 cards.
   Events: $A =$ drawing a King, $B =$ drawing a Queen.
   $A \cap B = \emptyset$ (no card can be both a King and a Queen)
4. Non-Mutually Exclusive Events: Two events are non-mutually exclusive if they can occur at the same time.
   Example:- Drawing a card from a deck and getting either a King or a Heart.
   Sample Space: S = all 52 cards.
   Events: $A =$ drawing a King, $B =$ drawing a Heart.
   One card (the King of Hearts) is both a King and a Heart, so $A \cap B \neq \emptyset$.
5. Independent Events: Two events are independent if the occurrence of one event does not affect the occurrence of the other.
   Example:- Flipping a coin and rolling a die.
   Sample Space for coin flip: $S_1 = \{H, T\}$
   Sample Space for die roll: $S_2 = \{1, 2, 3, 4, 5, 6\}$
   Events: $A =$ getting Heads, $B =$ rolling a 3.
6. Dependent Events: Two events are dependent if the occurrence of one event affects the occurrence of the other.
   Example:- Drawing two cards from a deck without replacement.
   Sample Space: S = all 52 cards.
   Event $A =$ drawing an Ace on the first draw.
   Event $B =$ drawing a King on the second draw.
   The probability of drawing a King on the second draw depends on whether an Ace was drawn first.
7. Complementary Events: The complement of an event $A$ is the event that $A$ does not occur.
   Example:- Rolling a die and not getting a 6.
   Sample Space: $S = \{1, 2, 3, 4, 5, 6\}$
   Event $A =$ rolling a 6.
   Complement of $A$, denoted $A'$ = not rolling a 6.
   $A' = \{1, 2, 3, 4, 5\}$
   $P(A') = 1 - P(A) = 1 - \frac{1}{6} = \frac{5}{6}$