Unit 3: Probability 3.1: Introduction to Probability

Definitions Fair game Experiment Trial All players have an equal chance of winning Each player can expect to win or lose the same number of times in the long run Experiment Has a well-defined outcome E.g. tossing a coin 20 times Drawing a card from a deck and replacing it 35 times Trial One repetition of an experiment E.g. flipping a coin once Drawing one card from a deck

Definitions Discrete random variable Expected value Simulation A variable that assumes a unique value for each outcome Expected value The value the mean of the random variable value tends towards after many repetitions Simulation An experiment that models an actual event Simple Event An event where there is only 1 outcome

Probabilities There are three kinds of probabilities Experimental Based on observation Also called empirical or relative frequency probability Theoretical Based on mathematical analysis Also called classical or a priori probability Subjective Estimate based on informed guesswork

Theoretical Probability Sample Space (S) Consists of all possible outcomes of an experiment Event Space (A) - Consists of all outcomes that correspond to the event of interest S A

Theoretical Probability n(S) The number of elements in set S n(A) The number of elements in set A The probability of an event A is

Theoretical Probability P(A) Likelihood that an event will occur 0 ≤ P(A) ≤ 1 P(A) = 0 Impossible event P(A) = 1 Event is a certainty P(A) < 0 or P(A) > 1 have no meaning Probability can be expressed as a percent, decimal, or fraction

Important Basic Info: Coins Coins have two faces Heads (H) Tails (T) A fair coin has equal likelihood of landing heads or tails

Important Basic Info: Dice Singular = die For a fair die, each side has equal likelihood of landing face-up A six-sided die has 6 sides: 1, 2, 3, 4, 5, 6 An eight-sided die has 8 sides: 1-8 A dodecahedral die has sides 12 Source: http://www.oitc.com/Dice/images/Dice.gif

Important Basic Info: Cards 52 cards in a deck 4 suits, 2 colours: spades ♠, clubs ♣, hearts ♥, diamonds ♦ 13 values in each suit: A(ce), 2, 3, 4, 5, 6, 7, 8, 9, 10, J(ack), Q(ueen), K(ing) Face cards: Jack, Queen, King Some decks may contain Jokers (2) Source: http://david.bellot.free.fr/svg-cards/images/svg-cards-2.0.jpg

Example 1: Rolling a Die Find the probability that a single roll of a die will result in a number less than 4. A = {1, 2, 3} Therefore the probability of rolling a number less than 4 is = 0.5

Example 2: Drawing a Card at Random A card is drawn at random from an ordinary deck of 52 playing cards. What is the probability of drawing a king? A = {K, K, K, K} Therefore the probability of drawing a king is = 0.077

Complementary Events S A' A The complement of an event A is given by A′ Means that the event does not occur E.g. A = die rolls a 3 A′ = die rolls anything other than a 3 A and A′ together will include all possible outcomes Sum of their probabilities must be 1 P(A) + P(A′) = 1 P(A′) = 1 – P(A) S A A'

Example 4 What is the probability that a randomly drawn integer between 1 and 40 is not a perfect square? A = set of perfect squares between 1 and 40 = {1, 4, 9, 16, 25, 36} A′ = not a perfect square Therefore the probability that a randomly drawn integer between 1 and 40 is not a perfect square is 0.85

#5a on HO Each of the letters of the word PROBABILITY is printed on same-sized pieces of paper and placed in a bag. The bag is shaken and one piece of paper is drawn. (Consider Y as a vowel.) A) What is the probability that the letter A is selected? A = letter A is drawn, S = number of letters there is 1 A and there are 11 letters Therefore the probability of drawing a letter A is = 0.09

#5b on HO What is the probability that the letter B is selected? A = letter B is drawn Therefore the probability of drawing a letter B is = 0.18

#5c on HO What is the probability that a vowel is selected? Therefore the probability of drawing a vowel is = 0.45

#5d on HO What is the probability that a consonant is not selected? A = consonant selected A′ = consonant not selected Note: this is the same as saying P(vowel selected) Therefore the probability of not drawing a consonant is 0.45.

FND: pg. 218 #1-15