Probability Week 5

Probability Definitions Probability – the measure of the likely hood of an event. Event – a desired outcome of an experiment. Outcome – a result from an experiment. Sample Space – Collection of all possible outcomes Equation: P(E) =

Probability Con’t All answers for probability questions: 0 < P(E) < 1 Thus all answers are written as fractions, decimals or percents, or 1 Zero! Certain Events = 1 or 100% Impossible Events = ZERO.

Classic Probability Experiments: Rolling a die There will be 6 outcomes in the sample space: {1, 2, 3, 4, 5, 6} Tossing a fair coin There will be 2 outcomes in the sample space: { Heads, Tails} Drawing a card from a standard deck There will be 52 cards in the sample space: {Spades: 2,3,4,5,6,7,8,9,10, ace, jack queen, king, Clubs: 2,3,4,5,6,7,8,9,10, ace, jack, queen, king, Diamonds: 2,3,4,5,6,7,8,9,10, ace, jack, queen, king, Hearts: 2,3,4,5,6,7,8,9,10, ace, jack, queen, king} Drawing one marble from the bottle There will be 8 marbles in the sample space: {blue marble, blue marble, blue marble, blue marble, blue marble, red marble, red marble, red marble}

Roll of a die = {1,2,3,4,5,6} 1)P(1) = 2)P(even) = 3)P(7) = 4)P(prime#) =

Bag ‘o’ marbles = {2 blue,3 red,5 white,1 yellow} 1) P(red) = 2)P(red or blue) = 3)P(green) = * Total of all probabilities = 1 (or 100%)

Deck of playing cards: {52 cards, 4 suits, red and black} 1)P(4 of diamonds) = 2)P(club) = 3)P(queen) = 4)P(face card) =

Compound Events: Involves two or more experiments at the same time. Ex. (1) Rolling two dice together: There will be 36 outcomes in the sample space: {(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)}

Compound Events con’t To find the total number of possibilities of more than one event, simply multiply the total number of possibilities of each. This is called the “Counting principle” Ex (1) Flipping 2 coins = 4, because each has two poss. Ex (2) The roll of a die and flip of a coin = 12 Ex (3) Selecting a outfit from 3 different shirts, 2 pairs of slacks, 5 ties = 3x2x5 = 30

Tree Diagram Tree diagrams are a visual outline (flow chart) of each possible event. When completed the will list all the possible outcomes of a Sample space. Ex – Flip of a coin and roll of a die:

With or Without Replacement Replacing simply means to place the object back into the total possibilities before re- drawing Without replacement means to keep the desired object out from the total. To find the probability of more than one event, Multiply their probabilities.

Deck of playing cards: (without replacement) {52 cards, 4 suits, red and black} 1)P(4 of diamonds and a 5 of clubs) = 2)P(two clubs) = 3)P(Ace and a King) = 4)P(four aces) =

Flipping coins Ex (1) What is the probability of getting Tails 3 times in a row? Ex (2) If we received tails 5 times in a row, what is the probability it will be tails on the 6 th toss? Ex (3) What is the probability of getting heads 6 times in a row?

Having kids? Ex (1) – If you were to have 4 children, what is the probability of having 4 boys? * Make a tree diagram to display the sample space.

How many ways? Ex (2) – At NCCC, there are 10 doors that can be used to enter the building and 2 stairways to the Library. How many different routes are there from outside the building to the Library, and then back out of the building?

Sandwich Shop: Ex (3) – A sandwich shop in town has Plain, Wheat and Italian rolls, 5 types of meat, 3 cheeses, and 4 toppings to choose from. (a)How many different sandwiches can be created with with one of each? (b)How many sandwiches can be created with a roll, two different meats, and cheese? (c) What is the probability of selecting a sandwich with a wheat roll?