PROBABILITY• RANDOM EXPERIMENTS• PROBABILITY OF

OUTCOMES• EVENTS• PROBABILITY OF EVENTS

RANDOM EXPERIMENTS• IF WE CANNOT FORSEE THE OUTCOME OF AN

EXPERIMENT IT IS SAID TO BE RANDOM

• SET OF POSSIBILITIES (UNIVERSAL SET) IS THE SET OF ALL POSSIBLE OUTCOMES REPRESENTED BY THE SYMBOL Ω

ex. Roll the die once - Ω = 1, 2, 3, 4, 5, 6

Venn Diagram

Ω

1 43

6 5 2

RANDOM EXPERIMENTS (CONT’D)

• TREE: STEP 1 STEP 2

HEADS …HEADS

TAILS …

HEADS …TAILS

TAILS …

RANDOM EXPERIMENTS (CONT’D)

• BASIC COUNTING RULE:• EXPERIMENTS INVOLVING 2 STEP:If the first step can be counted in a ways and the second step can be accomplished in b ways, then the total number of possible outcomes for this experiment is equal to the product

a x b• EXPERIMENTS INVOLVING 3 STEPS:If the third step can be accomplished in c ways then

a x b x c• Ex. If Eric has 2 pairs of pants (one black, one grey), 3

shirts (one white, one blue, one red), and 2 ties (one yellow, one red)

2 x 3 x 2 = 12 possible combinationsC:\Users\cobrien\Documents\PROBABILITY 10.notebook

PROBABILITY OF AN OUTCOME• Probability of an outcome in a random experiment

is equal to the number of FAVORABLE outcomes to the total number of POSSIBLE outcomes.

• P =

• Ex. Probability of flipping “Tails” =

• The sum of the probabilities = 1 (the closer to 1 the “more likely” a favorable outcome will occur)

PROBABILITY OF AN OUTCOME• EQUALLY PROBABLE SPACEWhen the possible outcomes of an experiment are “equally probable”, the set Ω is called EQUALLY PROBABLE SPACE.

Ex. The probability of pulling a “Heart”, “Spade”, “Diamond”, or “Club” 0r

• If Ω contains n possible outcomes, the probability of each outcome is

PROBABILITY OF AN OUTCOME• EXPERIMENT HAS SEVERAL STEPSEx. A dresser has 4 pairs of socks (3 white, 1 black)

REPLACEMENTSituation #1: 1st drawing 2nd drawing Ω outcome probability

W (W, W) x = W

B (W, B) x =

W (B, W) x = B

B (B, B) x =

PROBABILITY OF AN OUTCOME• 4 SOCKS (3 WHITE, 1 BLACK) NO

REPLACEMENTSituation #2: 1st drawing 2nd drawing Ω outcome probability

W (W, W) x = W

B (W, B) x =

BW (B, W) x =

EVENTS• An event is a “SUB SET” of the set Ω of

possibilities of a random experiment.Ex. Rolling only an even number A = (2, 4, 6)

• A certain event which always occurs and is noted Ω

• An impossible event which never occurs and is noted

• Any event of Ω containing only one element is called an ELEMENTARY EVENT.

EVENTS• INCOMPATIBLE EVENTS – mutually exclusive

events that cannot occur simultaneously.Ex. You cannot draw both a Spade and a Heart

• COMPLIMENTARY EVENTS – (contrary) if two events are incompatible AND if it is certain that either of these events will occur

Ex. Rolling an ODD number or rolling an EVEN number.

PROBABILITY OF AN EVENT• The probability of an Event (E) to occur is equal to

the SUM of the probabilities of favorable outcomes.

Ω = (a, b)P(E) = P(a) + P(b) What is the probability of spinning anEVEN number? Ω (2, 4, 10) + + =

What is the probability of spinning an 8? Ω (10)

1

2

34

25 3

10

PROBABILITY OF AN EVENT• DEPENDENT AND INDEPENDENT EVENTS

• INDPENDENT – occurs when one event does not influence the probability of the other occurring.

Ex. Pulling a card from the deck and replacing it. You are choosing from a 52 card deck every time.

• DEPENDENT – occurs when one event influences the probability of the other occurring.

Ex. Pulling a card from a deck and NOT replacing it. You are choosing from a decreasing number of cards each time.