PROBABILITY

The Probability Scale

Probability measures how likely an event is to happen. It is always a value between 0 and 1.
    * 0 = Impossible
    * 0.5 = Even Chance
    * 1 = Certain
            Probability is described in FRACTION, DECIMAL or PERCENTAGE

Basic Formula

        $P(Event) = \frac{(Number \;of\; Favourable \;Outcomes)}  {(Total \;Number \;of\; Possible\; Outcomes)}$

    Single event probability 
        *A single event is a one-step process, like tossing a coin once or picking one marble from a bag.
        Key Requirement: You must identify the "Sample Space" (the list of all possible results).
        *The "Sum of 1" Rule: The sum of the probabilities of all possible single events 

        in a sample space must equal 1.
        *Complementary Events: If the probability of winning is 0.2, 

        the probability of not winning is 1-0.2 = 0.8.

    Experimental probability (Relative Frequency)

        P(A) = $\frac{number \;of\; time\; A\; occurs}{total\; number\; of\; trials}$

        More trials ----- more accurate 

        $Relative frequency = \frac{number\; of\; success}{total\; number\; of\; trials}$

    Theoretical probability 

        $P(A)=\frac{no \;of\; desired \;outcome}{total \;no\; of\; possible \;outcome}$

While theorectical probability tells us what should happen, Experimental probability tells us what actually happened during a test.

The "NOT" Rule (Complementary Events)

    P(A)+P(not A) = 1
    The probability of something NOT happening is:
    P(Not A) = 1 - P(A)
    Example: If the probability of rain is 0.3, the probability of it NOT raining is 1 - 0.3 = 0.7

Combined Events

    * The "OR" Rule (Addition): For mutually exclusive events, P(A or B) = P(A) + P(B).
    * The "AND" Rule (Multiplication): For independent events, P(A and B) = P(A) * P(B).

Tree Diagrams

    Tree diagrams help visualize multiple stages of an event.
        * Branches: Multiply probabilities along the branches.
        * Ends: Add the probabilities at the ends of the branches if you need multiple outcomes.
        * Important: Always check if the total number of items decreases (Without Replacement). ie                       denominator

Venn Diagrams & Sets

    * Intersection (A ∩ B): The middle part where both events happen (A AND B).
    * Union (A ∪ B): Everything inside both circles (A OR B).
    * Complement (A'): Everything outside circle A.

When solving "Without Replacement" problems, remember to subtract 1 from both the numerator and the denominator for the second selection!P(A) = probability of A happening