Probability Seeing structure and order within chaotic, chance events. Defining the boundaries between what is mere chance and what probably is not. Asymptotic Trend : However, we never reach a stable exact 50/50 proportion with a finite (limited) number of tosses. In fact, as the number of tosses increases, the probability of an exact 50/50 proportion decreases and approaches zero. Coin toss example: As you increase the number of tosses (of a coin), the gap between the observed proportions and the expected proportions (50/50) closes, but by progressively smaller amounts.

Probability as Rationality raised to Mathematical Precision Probability as a ratio: If there is one white ball and three black balls in a bag and you reach in and draw one out, which colour would you bet that you picked? There is one chance in four of picking the white ball. There are three chances in four of picking a black ball. If we pick a ball out of the bag a large number of times (replacing each time), what percentage of the time would we select a black ball? (Assuming that we don’t know what is in the bag.) Probability describes the structure that exists within a population of events. Relative Frequency: A Priori vs. A Posteriori probability A probability equals the ratio of the number of possibilities favorable for the event over (divided by) the total number of possible events.

Flipping a coin six times. Number of heads tossed What proportion of them will have tossed all heads or all tails? A large number of people toss a coin six times: What proportion of them will have tossed three heads and three tails?

Emerging pattern from coin tossing 1.While the details may differ, the distribution will be symmetrically arranged around the central values. 2. The most frequently occurring values are the central one. 3. The least frequently occurring values lie the farthest distance from the centre. 4. The relative frequencies of the intermediate values decreases in a regular and symmetrical fashion as we move from the centre to the periphery.

Probability: Another Analysis Analytic View: If an event can occur in A number of ways, and if it can fail to occur in B ways, then P(event) equals A divided by A+B. Example: event rolling an even number on a die. A = P(event) =3/(3+3) = ½ = 0.5 Relative Frequency View: Subjective View: Belief in the likelihood of an event. Read as “the probability of”….what ever is in the brackets. 3 {2,4,6} 3 {1,2,5}B = empirical sampling with replacement probability as a limit of relative frequency rational logical expected frequency

Terminology Event: that with which we are concerned Independence: Example: bluebrown m30 f20 bluebrown m30 f1030 In both cases, there are 100 observations IndependentNot Independent two events are independent when the occurrence of one does NOT influence the P (probability) of the other occurring.

Mutually Exclusive: Example: One is either a man or a woman. Being one precludes the other. Exhaustive: A set of events is exhaustive if the set includes ALL possible outcomes. Example: roll of the die (1,2,3,4,5,6) Probability can range from 0.0  1.0. Two events are mutually exclusive if the occurrence of one event precludes the occurrence of the other.

Joint Probabilities: The probability of the co-occurrence of two or more events, if they are independent, is given as… Example: Conditional Probabilities: BlueBrown Males4060 Females6040 / = if, or given, this event has occurred. Are gender and eye colour independent? Eye Colour

Probability Distribution of Discrete Variables You can ask………. What is the P of a 3? Where do the fractions in the formula come from?

Laws of Probability Disjunctive: (A or B) Conjunctive: (A and B) Or P(A)P(B) Conjunctive: Multiplicative Law P(two head in two tosses of a coin) ABAB T TTTHHTHHTTTHHTHHH All possible pairs of the events numerator is the number of favorable events Restriction: all events must be independent What always works is, P(H/A and H/B) = P(H/A) * P(H/B if there was a H/A) denominator is the total number of possible events

Disjunctive: Additive Law Tossing a die: The probability of tossing a 1 or tossing a 3 is equal to the sum of the probabilities of the two separate events, i.e.: 1/6 + 1/6 = 2/6 =.333 Restriction: The events must be mutually exclusive. ABAB T TTTHHTHHTTTHHTHHH Given two tosses (A and B): NOT ½ + ½ = 1 P(H/A or H/B) = P(H/A) + P(H/B) – P(H/A and H/B) = P(H/A and H/B) is the product of the probabilities of the two events. See previous page. P(1)P(3)P(1 or 3)

More than two events. P(H/A) or P(H/B) or P(H/c) Convert to P(not H/A)…. and, use multiplicative law, and, subtract the product from P(T/A) and (T/B) and (T/C) = 1 -.5(.5)(.5) =.875 Thus: P(H/A) or P(H/B) or P(H/C) =.875 Given P(H) =.5 and P(T) =.5)