3.1– 3.4 Events, Sample Space, Probability, and their Properties Key concepts: events, sample space, probability Probability is a branch of mathematics that deals with modeling of random phenomena or experiments, that is experiments whose outcomes may vary when repeated Examples: 1. A toss of a coin 2. A roll of a die 3. A random selection of an individual from a population Intuitively probability of an event is a long-run proportion of times the event occurs in in many independent repetitions of the experiment. # of times A has occured P( A) Law of Large Numbers: # of repetition s Model of a random phenomenon: the sample space, events, and probability Example 1. A coin is tossed three times. Find the probabilities of the following events: 1. Exactly two “heads” (H) in three tosses 2. “Heads” in first toss 3. Exactly two “heads” in three tosses AND “heads” in the first toss 4. Exactly two “heads” in three tosses OR “heads” in the first toss SOLUTION: The sample space HHH HHT HTH THH HTT THT TTH TTT S= Events: A = "exactly two H in three tosses" = {HHT, HTH, THH} B = "H in first toss" = {HHH, HHT, HTH, HTT} Probability: assume that the coin is fair and hence each sample point has the same probability P(a single sample point) = 1/8 1. 2. 3. 4. (equally likely outcomes) P(exactly two H in three tosses) = P(A) = 3/8 P(H in the first toss) = P(B) = 4/8 = 1/2 P(“exactly two H in three tosses” AND “H in the first toss”) = P(A ∩ B) = = P({HHT, HTH} ) = 2/8 = 1/4 P(“exactly two H in three tosses” OR “H in the first toss”) = P(A B) = = P{ HHH, HHT, HTH, HTT, THH}) = 5/8 The most basic outcome of an experiment is called a sample point. Venn Diagram 1. The sample space S = the set of all possible sample points 2. Events (marked A, B..) = subsets of the sample space. We say that an event A occurs if an outcome is in the set A. S A The union A B of two events A and B is the event that occurs if either A or B or both occur (on a single performance of the experiment.) A B consists of all the sample points that belong to A or B or both The intersection A ∩ B of two events A and B is the event that occurs if both A and B occur at the same time. A ∩ B consists of all the sample points belonging to both A and B. The complement Ac of an event A is the event that A does not occur, that is, the event consisting of all sample points that are not in A. A AC 3. Probability = a measure of the likelihood of an event measured by a number between 0 and 1. Mathematically, probability is a function that to each event A assigns a number P(A) (called the probability of an event A) Properties of probability: 0 P(A) 1 P(S) = 1 P(Ac) + P(A) = 1 [Rule of Complements] in particular P(A) = 1- P(Ac) and P(Ac) = 1- P(A) P(A B) = P(A) + P(B) - P(A ∩ B) [Additive Rule] A A∩B B Events A and B are mutually exclusive if A ∩ B contains no sample points, that is, if A and B cannot occur at the same time. For mutually exclusive events A and B P(A B) = P(A) + P(B) Example 1 (cont). A coin is tossed three times. Find the probability of: 1. Exactly two H in three tosses OR H in the first toss using the Additive Rule 2. At least two H in three tosses 3. At most one H in three tosses using the Rule of Complements SOLUTION 1. P(“exactly two H in three tosses” OR “H in the first toss”) = P(A B) = P(A) + P(B) - P(A ∩ B) = 3/8 + 4/8 – 2/8 = 5/8 Let D = “at least two H” = {HHH, HHT, HTH, THH} HHH HHT HTH THH HTT TTH TTT 2. P(D) = 4/2 = 1/2 THT 3. P(most one H) = 1 – P(least two H ) = 1 – P(D) = 1-1/2 = 1/2 Equally Likely Outcomes Model. Assume that there are finitely many possible outcomes and all are equally likely. Then P( A) n( A) # of outcomes in A n( S ) # of outcomes in S Simple Random Sample. If n elements are selected from a finite population is such a way that every set of n elements has an equal chance of being selected, then the n elements are said to be a simple random sample (SRS). Combination Rule assumes that sample is taken without replacement and without regarding the order (order in which elements are selected does not matter). Example 1. A “poker hand” consists of 5 cards selected from a standard deck of 52 cards , order does not matter. The number of different “poker hands” is TI-83: 52 – MATH – PRB –3:nCr – ENTER – 5 – ENTER Exercises. 1. In Florida’s state lottery game, called Pick-6 Lotto, you select six numbers of your choice from a set of numbers ranging from 1 to 53. What is the probability of winning Lotto? 2. Two fair dice are tossed, and the face on each die is observed. a. Assign probabilities to each sample point b. Find the probability of each of the following events: A = {3 showing on each die} B = {sum of two numbers showing is 7} C = {sum of two numbers showing is even} 3. [3.30 p. 151] A pair of fair dice is tossed. Define the following events: A = {You will roll a 7} (i.e., the sum of the dots on the two dice is equal to 7) B = {At least one of the two dice shows a 4} a. Identify the sample points in the events A, B, A ∩ B, A B, Ac Find P(A) = …….., P(B) = ………., P(Ac ) = …….. P(A ∩ B) = ………, and P(A B) = …….., by summing the probabilities of sample points b. Find P(A B) using the additive rule P(A B) = …………………………………… c. Are A and B mutually exclusive? Why 4. Two marbles are drawn at random and without replacement from a box containing four blue marbles and two red marbles. a. List the sample points for this experiment. b. Assign probabilities to the sample points (make the probability tree). c. Determine the probability of observing each of the following events: A: {Two blue marbles are drawn.} B: {A red and a blue marble are drawn.} 5. Consider the Venn diagram below, where P(E1) = .10, P(E2) = .05, P(E3) = P(E4) = .2, P(E5) = .06, P(E6) = .3, P(E7) = .06, and P(E8) = .03. Find the following probabilities: a. P(Ac) = ………….. b. P(Bc) = ………….. e. P(Ac ∩B) = ………….. d. P(A B) = ………….. e. P(A∩B) = ………….. f. P(Ac∩Bc) = ………….. g. Are events A and B mutually exclusive? Why? 6. (Ex. 3.11, p. 147) Consider the experiment of tossing fair coins. Define the following event: A: {Observing at least one head}. a. Find P(A) if 2 coins are tossed. b. Find P(A) if 10 coins are tossed.

© Copyright 2020