2017 Fall QMS102 Tip Sheet 2

Transcription

1 Chapter 5: Basic Probability 2017 Fall QMS102 Tip Sheet 2 (Covering Chapters 5 to 8) EVENTS -- Each possible outcome of a variable is an event, including 3 types. 1. Simple event = Described by a single characteristic. (e.g. A day in January from all days in 2017) 2. Joint event = Described by two or more characteristics. (e.g. A day in January that is also a Wednesday from all days in 2017) 3. Complement of an event A (denoted by A ) = All events that not part of event A. (e.g. All days from 2012 that are not in January) SAMPLE SPACE the collection of all possible events. (e.g. All 6 faces of a dice) To Visualize Events: Contingency Tables Decision Trees MUTUALLY EXCLUSIVE EVENTS Events that cannot occur simultaneously. (e.g. randomly choosing a day from A = day in January. B = day in February. Events A and B are mutually exclusive) Probability: a measure of the likelihood that event X will happen, denoted by P(X). - All probabilities are between, event will not happen 0 P(X) 1, event will surely happen. P(X) = (X N) * 100% - The sum of all probabilities in the sample space is 100% = 1 - Chance is expressed in percentage % (E.g. 50% chance of getting tail) - Probability of X not happening = 1 P(X) (e.g. If P (raining) = 2/3 = 66%, P (Not raining) = 1 (2/3) = 33% ) The probability of a joint event, A and B P (A&B) = ( # of outcomes satisfying both A and B) Total # of Outcomes The probability of a marginal or simple event P (A) = P (A and B1) + P (A and B2) + P (A and B3) + ```+ P (A and Bk), where B 1, B 2,, B k are k mutually exclusive and collectively exhaustive events. Page 1 of 6

2 General Addition Rule: P(A or B) = P(A) + P(B) P(A and B), if A and B are mutually exclusive, then P (A and B) = 0, so we get P(A or B) = P(A) + P(B) When two events are independent ONLY if P(A B) = P(A) ---Event A and B are independent when the probability of one event is not affected by the fact that the other event has occurred. Multiplication Rule Multiplication rule for two events A and B: P(A and B) = P(A B) x P(B) If A and B are independent, then P(A B) = P(A) and the multiplication rule simplifies to P(A and B) = P(A) x P(B) Conditional probability is the probability of one event happening, given that another event has occurred already. The conditional probability of A given B (has occurred): P(A B) = P (A&B) P(B) The conditional probability of B given A (has occurred): P(B A) = P (A&B) P(A) Counting Rule 1 If any one of k different mutually exclusive and collectively exhaustive events can occur on each of n trials, the number of possible outcomes = k n. Page 2 of 6

3 Counting Rule 2 If there are k1 events on the 1 st trail, k2 events on the 2 nd trail,, and kn events on the n th trail, then the number of possible outcomes = k1 k2 kn Counting Rule 3 The number of ways that all n items can be arranged in order is n! (called n factorial), where n! = n (n-1) (n-2) (n-3) 1, and 0! =1 Counting Rule 4 Permutations The number of ways of arranging x objects selected from n objects in order is Counting Rule 5 Combinations npx = n! / (n x)! The number of ways of selecting x objects from n objects, irrespective of order, is ncx = n! / [x! (n - x)!] Chapter 6: Discrete Probability Distribution A random variable represents a possible numerical value from an uncertain event. Discrete Random Variables produce outcomes that come from a counting process (e.g. number of courses you are taking). They can only assume a countable number of values. (e.g. rolling a die twice. If X was the number of times a 4 can occur, X can be 0, 1, or 2.) Discrete Probability Distribution A table of all possible numerical outcomes for that variable and a probability of occurrence associated with each outcome. X variable Frequency Relative Frequency P(X) X1 1 1/4 = 25% X2 2 2/4 = 50% X3 1 1/4 = 25% Note: Use CALC function on Calculator to compute the Expected Value ( x ), Variance, and Standard Deviation. Generally, the higher the expected value, the better the option is. The lower the standard deviation, the less risky the option is. For a population CV = (σ / µ) * 100%, for a sample CV = (s / x ) * 100% Page 3 of 6

4 Binomial Values: Finite number of values. The possible value ranges from 0 to n, given the experiment which has n trails, each trail has 2 clear outcomes, success or failure. P (success) = x / n = π P (failure) = (n-x) / n = 1 - π Binomial Distribution: Mean: Variance: Standard Deviation: Poisson Values: Infinite number of values. The possible value ranges from 0 to, given an experiment (e.g. how many cars will pass by the intersection between 12 pm and 2pm?) It measures/analyzes # of events of interest occurred during a period of time or amount of space (area, volume, length, area, height). Success will occur randomly, and it must be independent of each other. The average rate of success = lambda (λ), where X = # of success in a certain amount of time or space. Poisoon Distribution: Mean: Variance: Standard Deviation: Calculator Instructions: STAT DIST (F5) BINM or POISN Bpd or Bcd or Ppd or Pcd Data: Variable (always) Input values x, n, p=probability given, or µ = λ EXE Note: Bpd = Binomial Probability Distribution, means P (X= #) Bcd = Binomial Cumulative Distribution, means P (X #) Ppd = Poisson Probability Distribution, means P (X= #) Pcd = Poisson Cumulative Distribution, means P (X #) For either Bcd or Pcd, we need to input the default statement P (X #) into the calculator. To translate what we get from the questions directly, we need to: Step 1, Convert words into signs, X more than A= greater than=bigger than= X > A X is no more than A= at most = less or equal to = X A X is at least A = greater or equal to = no less than = X A X is less than A = fewer than = smaller than = X < A Step 2 Transform into default statement that calculator can proceed. Page 4 of 6

5 P (X>A) = 1 P (X A); P (X<A) = P (X A-1); P (X A) = 1 P(X A-1) P ( B X A) = P (X A) P (X B-1) Chapter 7 The Normal Distribution Continuous Normal Distribution Properties: Symmetrical (mean = median). Total area is 1 = 100% Bell-shaped (Most values are clustered around the middle) With infinite range from - to To find P ( X A) =? To find the probability, use Normal Cumulative Distribution ( Ncd ) function on the calculator : STAT DIST NORM Ncd Data: Variable ( always) Lower: minimum value given or 100,000 Upper: Maximum value given or 100, σ: standard deviation -- μ: mean EXE (See the previous 2 Step Translation if needed) Given P (X?) = B, to find the answer, use Inverse Normal ( InvN ) function on the calculator: STAT DIST NORM InvN Data: Variable ( always) Tail : Left / Right / Centre Area: B (the given probability) -- σ: standard deviation -- μ: mean EXE (See the previous 2 Step Translation if needed) To decide the Tail choice, smallest, lowest, bottom, less than, fewer than Left Greater than, above, exceed, top, more than, largest Right Between A and B CNTR Z score: Using the transformation formula, normal distributions can be transformed to its standardized form Z, also known as the z score. Z = x µ x x = σ s A standarized Z value has a mean µ = 0 and standard deviation σ = 1, which can be used to compute the probability. Standard Normal Distribution According to the Empirical Rule, all normal distribution curve follows the same rule. Page 5 of 6

6 Chapter 8 Sampling and Sampling Distribution Using Samples is LESS time consuming, LESS costly to collect, and MORE PRACTICAL to test and analyze. Sampling distribution the distribution of the results if you actually selected all possible samples. Sample mean is unbiased: sample mean µ x = µ population mean Standard Error of the Mean: standard deviation of all sample means σ x = σ n Finding Z for the Sampling Distribution of the Mean, Z = x µ x µ x Finding x for the Sampling Distribution of the Mean, x = µ + Z σ Π (1 Π ) Standard Error of the Proportion: σ p = n Finding Z for the Sampling Distribution of the Proportion, Z = ( Π is the population proportion, and the sample proportion is p = x n ) Central Limit Theorem n = x µ σ n p Π Π (1 Π ) n states that as the sample size gets larger enough, (greater than 30 regardless of the shape, 5 for fairly symmetrical, regardless of the sample size for normally distributed), the sampling distribution of the mean is approximately normally distributed, regardless of the distribution of the individual values in the population. Page 6 of 6