Probability

Random Experiment… …a random experiment is an action or process that leads to one of several possible outcomes. For example: Experiment Outcomes Flip a coin Heads, Tails Exam Marks Numbers: 0, 1, 2, ..., 100 Assembly Time t > 0 seconds Course Grades F, D, C, B, A, A+

Probabilities… List the outcomes of a random experiment… List: “Called the Sample Space” Outcomes: “Called the Simple Events” This list must be exhaustive, i.e. ALL possible outcomes included. Die roll {1,2,3,4,5} Die roll {1,2,3,4,5,6} The list must be mutually exclusive, i.e. no two outcomes can occur at the same time: Die roll {odd number or even number} Die roll{ number less than 4 or even number}

Sample Space… S = {O1, O2, …, Ok} A list of exhaustive [don’t leave anything out] and mutually exclusive outcomes [impossible for 2 different events to occur in the same experiment] is called a sample space and is denoted by S. The outcomes are denoted by O1, O2, …, Ok Using notation from set theory, we can represent the sample space and its outcomes as: S = {O1, O2, …, Ok}

Requirements of Probabilities… Given a sample space S = {O1, O2, …, Ok}, the probabilities assigned to the outcome must satisfy these requirements: The probability of any outcome is between 0 and 1 i.e. 0 ≤ P(Oi) ≤ 1 for each i, and The sum of the probabilities of all the outcomes equals 1 i.e. P(O1) + P(O2) + … + P(Ok) = 1

Approaches to Assigning Probabilities… There are three ways to assign a probability, P(Oi), to an outcome, Oi, namely: Classical approach: make certain assumptions (such as equally likely, independence) about situation. Relative frequency: assigning probabilities based on experimentation or historical data. Subjective approach: Assigning probabilities based on the assignor’s judgment. [Bayesian]

Classical Approach… If an experiment has n possible outcomes [all equally likely to occur], this method would assign a probability of 1/n to each outcome. Experiment: Rolling a die Sample Space: S = {1, 2, 3, 4, 5, 6} Probabilities: Each sample point has a 1/6 chance of occurring. What about randomly selecting a student and observing their gender? S = {Male, Female} Are these probabilities ½?

Classical Approach… Experiment: Rolling 2 die [dice] and summing 2 numbers on top. Sample Space: S = {2, 3, …, 12} Probability Examples: P(2) = 1/36 P(6) = 5/36 P(10) = 3/36 1 2 3 4 5 6 7 8 9 10 11 12

basic concepts probability of event = p .5 = even odds 0 = certain non-occurrence 1 = certain occurrence .5 = even odds .1 = 1 chance out of 10

basic concepts (cont.) if A and B are mutually exclusive events: P(A or B) = P(A) + P(B) ex., die roll: P(1 or 6) = 1/6 + 1/6 = .33 possibility set: sum of all possible outcomes ~A = anything other than A P(A or ~A) = P(A) + P(~A) = 1

basic concepts (cont.) discrete vs. continuous probabilities discrete finite number of outcomes continuous outcomes vary along continuous scale

discrete probabilities .5 p .25 HH HT TT

continuous probabilities .1 .2 p .1 .2 p total area under curve = 1

independent events one event has no influence on the outcome of another event if events A & B are independent then P(A&B) = P(A)*P(B) if P(A&B) = P(A)*P(B) then events A & B are independent coin flipping if P(H) = P(T) = .5 then P(HTHTH) = P(HHHHH) = .5*.5*.5*.5*.5 = .55 = .03

if you are flipping a coin and it has already come up heads 6 times in a row, what are the odds of an 7th head? .5 note that P(10H) < > P(4H,6T) lots of ways to achieve the 2nd result (therefore much more probable)

conditional probability Conditional probability is used to determine how two events are related; that is, we can determine the probability of one event given the occurrence of another related event. Experiment: random select one student in class. P(randomly selected student is male) = P(randomly selected student is male/student is on 3rd row) = Conditional probabilities are written as P(A | B) and read as “the probability of A given B” and is calculated as:

Conditional Probability… Again, the probability of an event given that another event has occurred is called a conditional probability… P( A and B) = P(A)*P(B/A) = P(B)*P(A/B) both are true Keep this in mind!

CONDITIONAL PROBABILITY In a class of 20 students 10 study French, 9 study Maths and 3 study both French Maths S 7 3 6 4 The probability they study Maths given that they study French The probability of P(MF) = P(MF) / P(F) = 3/10

conditional probability … P(B|A) = P(A&B)/P(A) if A and B are independent, then P(B|A) = P(A)*P(B)/P(A) P(B|A) = P(B) One of the objectives of calculating conditional probability is to determine whether two events are related. In particular, we would like to know whether they are independent, that is, if the probability of one event is not affected by the occurrence of the other event.