Unit 0: Probability

PROBABILITY— how likely an event is to occur

“random” is a description of a kind of pattern that emerges in the long-run in a chance phenomena

Law of Large Numbers: after many repetitions of any chance process, the relative frequency of a particular outcome approaches the actual probability of that outcome

Combining these two concepts…

probability: the proportion of times an outcome of a random phenomena would occur in a very long series of reptition; long-term relative frequency

0≤P(A)≤10\le P\left(A\right)\le1

    probability model: all possibile outcomes/their possibilities

sample space: the set of all possible outcomes for a random phenomenon

    event: a subset of the sample space that is any outcome of a random phenomenon

PROBABILITY RULES

  1. Probability is a number between 0 and 1

    0≤P(A)≤10\le P\left(A\right)\le1

    Probability 0 - the event is certain to not occur

    Probability 1 - the event is certain to occur

  2. All possible outcomes have a combined probability of 1

    P(S)=1P\left(S\right)=1

  3. The complement of any event is the probability of that event not happening, ACA^{C}

    The Complement Rule: P(AC)=1−P(A)P\left(A^{C}\right)=1-P\left(A\right)

ONE EVENT

  1. Use addition rule for disjoint events (“mutually exclusive”) with no common outcomes (can never occur simultaneously)

    The Addition Rule: P(A∪B)=P(A)+P(B)P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)

    Use general addition rule for events that are joint (“mutually inclusive”) with common outcomes (can occur simultaneously)

    The General Addition Rule: P(A∪B)=P(A)+P(B)−P(A∩B)P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)

MULTIPLE EVENTS

  1. Use multiplication rule for multiple, independent events (the outcome of the first event does not affect the outcomes of subsequent events)

    The Multiplication Rule: P(A∩B)=P(A)⋅P(B)P\left(A\cap B\right)=P\left(A\right)\cdot P\left(B\right)

  2. Use general multiplication rule for multiple, dependent events (the outcome of the first event affects the outcomes of subsequent events)

    The General Multiplication Rule: P(A∩B)=P(A)⋅P(B∣A)P\left(A\cap B\right)=P\left(A\right)\cdot P\left(B\vert A\right)

    The Conditional Probability: P(B∣A)=P(A∩B)P(B)P\left(B\vert A\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)}

    Tree diagram, Venn diagram, Two-way table

    tree diagram: use with probabilities with mutually exclusive events    

    venn diagram: use with probabilities involving unions and intersections

    union: this event OR that event; A∪BA\cup B

    intersection: this event AND that event; A∩BA\cap B

    complement: all elements NOT in that group; ACA^{C}

two-way tables: use with multiple variables/categories

COMBINATIONS

combinations: the number of ways an event can occur

Combinations: C(n,r)=n!r!(n−r)!C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}

    nn is the total number of objects in the set

    rr is the number of choosing objects from the set

TI-84: [MATH] > [PROB] > 3:nCr > nCr_{n}C_{r}

Manually: Multiply the product of the probabilities of one way by the number of ways the even can occur

“At least 1” = “1 or more” = complement of none; 1−P(none)1-P\left(none\right)

Multiple events within multiple events: Multiply the probability of each event with the probabilities of two eventsb.