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RANDOM EXPERIMENT STATS 103.3 - a process of observation such that - there is more than one possible observation, and - the observation that occurs cannot be predicted with certainty. PROBABILITY Definitions Models for Probability 1 2 OUTCOME or ELEMENT Random Experiment • one of the possible observations when a random experiment is performed. • May denote such an element as e e.g. A card is selected at random from a standard deck of cards. The card selected could be any one of the 52 cards in the deck. It is impossible to predict with certainty which card will be obtained. • e.g. Each of the 52 cards in the deck is an element or outcome. e1 = A♠ (Ace of Spades is observed) e2 = K♦ (King of Diamonds is observed) 3 4 STANDARD DECK OF 52 CARDS SAMPLE SPACE • the set of all possible outcomes of a random experiment. (denote it S ) • S = { e1, e2, e3, … eN } • e.g. S = {the 52 cards in a deck} • When a coin is tossed, S = { H, T } 5 A♣ A♦ A♥ A♠ 2♣ 2♦ 2♥ 2♠ 3♣ 3♦ 3♥ 3♠ 4♣ 4♦ 4♥ 4♠ 5♣ 5♦ 5♥ 5♠ 6♣ 6♦ 6♥ 6♠ 7♣ 7♦ 7♥ 7♠ 8♣ 8♦ 8♥ 8♠ 9♣ 9♦ 9♥ 9♠ 10 ♣ 10 ♦ 10 ♥ 10 ♠ J♣ J♦ J♥ J♠ Q♣ Q♦ Q♥ Q♠ K♣ K♦ K♥ K♠ 6 1 EVENT TRIAL • one repetition of a simple experiment. • a subset of the sample space. –Denote events as E, A etc. • Examples: – Selecting one card. A second trial is then the selection of another card. – Rolling a die. A second trial is a second roll. – Toss a coin. Toss it again. • e.g. H = {card selected is a Heart} M = {number on die is a multiple of 3} = { 3, 6 } 7 8 SIMPLE EVENT • an event involving one single outcome or element of the sample space. • Thus, E = { e } e.g. A = {card is the Seven of Hearts} = { 7♥} B = { Heads } 9 10 CERTAIN or SURE Event - This event MUST happen when the experiment is performed. - It is equivalent to the whole sample space S - e.g. E = {card is red or black} = S L = {number on die is less than 10} = S 11 12 2 IMPOSSIBLE or NULL Event What is PROBABILITY? - This event CANNOT happen when the experiment is performed. PROBABILITY is the likelihood or chance of the occurrence of an event or outcome. - It is denoted φ - It contains NO elements of the sample space S e.g. One card is selected at random from a standard deck of 52 cards. What is the probability that an Ace is chosen? - e.g. G = {card is green} = φ N = {number on die is negative} = φ 13 14 BASIC PROPERTIES NOTATION If E is an event in a sample space S, then the probability of the event E is denoted P[ E ] or Pr[ E ] 1. 0 ≤ P [ E ] ≤ 1 e.g. P[ F ] could denote the probability of obtaining a Face card when one card is drawn at random from a deck. 3. For any two distinct elements e and f of S, 2. for every event E. (probability is a number between zero and one) P [S ] = 1 (The probability of the sure event is one. The total probability available is one.) P [ e or f ] = P [ e ] + P [ f ] (for distinct outcomes, probabilities add) 15 Models for Probability 16 Examples 1. EQUALLY LIKELY or Classical Probability 1. What is the probability that a card drawn at random from a standard deck is a Face card? #( F ) 12 3 P [F] = = = = 0.2308 # ( S) 52 13 If the sample space S of a random experiment contains N = # (S) equally likely outcomes, and if an event E contains n = # (E) of these outcomes, then # (E ) n P [E ] = = # (S ) N 2. What is the probability that a multiple of 3 is observed when a die is rolled? #( M) 2 1 P[ M] = = = = 0.3333 #( S) 6 3 17 18 3 Models for Probability Models for Probability 2. RELATIVE FREQUENCY Probability 2. RELATIVE FREQUENCY Probability If a random experiment is repeated for M trials, and if this results in m occurrences of an event E, the relative frequency of occurrence of E is f = The probability of the event E is the limiting value of the relative frequency as the number of trials M increases without bound (goes to infinity). That is m M P [E ] = {} lim m M→∞ M 19 Examples 20 Models for Probability 1. What is the probability that a thumbtack will fall with its point “up” when dropped on the floor? 2. What is the probability that a skilled dart player will be able to hit the region of the dart board at which he/she aims? 3. EMPIRICAL (or Statistical) Probability The probability of the event E is approximated by the relative frequency of its occurrence based on a number of trials of a random experiment. P [E ] ≅ m M 21 Examples Models for Probability 1. Students in a certain class were classified according to gender and eye colour. Female Male Blue 28 22 Brown 36 33 Hazel 17 5 4. SUBJECTIVE Probability Other 9 10 After analyzing and studying all available information relevant to the problem, personal judgment is used. 2. Lengths in cm of fish caught in a lake last summer were summarized as follows. Length Number < 20 72 20 – 40 124 40 – 60 156 22 > 60 48 e.g. What is the probability of snow tomorrow? 23 24 4 Models for Probability 5. ODDS and Probability Odds indicate the number of equally likely chances for an event happening versus the number of equally likely chances against it happning. If the odds in favour of an event E occurring are m:n, then the probability of event E happening is m m+n e.g. The probability of a student successfully completing STATS 103.3 is P [E ] = P [ E] = e.g. The odds are 7:2 that a student will successfully complete STATS 103.3 . m 7 7 = = = 0.7778 m+ n 7 +2 9 25 If the odds against an event E occurring are m:n, then the probability of event E happening is n P [E ] = m+n e.g. In a race, the odds against a certain horse are quoted as 5:3 . The indicated probability of this horse winning the race is n 3 3 P [E] = = = = 0.3750 m+ n 5 + 3 8 26 Probability to Odds If the probability of an event E occurring is P[ E ], then the odds in favour of E are determined by starting with the ratio C P [ E ] : P E and reducing it to the form m:n in which m and n are the smallest possible positive integers having no common factors. 27 28 Example: The probability of a certain Huskie team winning its next home game is 0.84 . What are the odds in favour of this happening? Probability to Odds If the probability of an event E occurring is P[ E ], then the odds in favour of E are determined by starting with the ratio C P [ E ] : P E C ⇒ 0.84 : 0.16 P [ E ] : P E ⇒ 84 : 16 ⇒ 21 : 4 The odds against are just the reverse ratio The odds against them winning are then 4 : 21 P E C : P [ E ] 29 30 5