![ 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}
6.6](https://image.slidesharecdn.com/9qsbx8vcrjanmxxdgkzw-signature-a4078f78569ff597de72b4bfa668ea6779f2fb21b7c17a22d8c7003b362b53f6-poli-161102164355/85/Probability-6-320.jpg)
Probability
- 2. Introduction to Probability, Conditional Probability and Random Variables June 3, 2008 2Stat 111 - Lecture 6 - Probability
- 3. In deterministic processes, the outcome can be predicted exactly in advance • Eg. Force = mass x acceleration. If we are given values for mass and acceleration, we exactly know the value of force In random processes, the outcome is not known exactly, but we can still describe the probability distribution of possible outcomes • Eg. 10 coin tosses: we don’t know exactly how many heads we will get, but we can calculate the probability of getting a certain number of heads June 3, 2008 3Stat 111 - Lecture 6 - Probability
- 4. …a random experiment is an action or process that leads to one of several possible outcomes. For example: 6.4 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+
- 5. 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} 6.5
- 6. 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} 6.6
- 7. Given a sample space S = {O1, O2, …, Ok}, the probabilities assigned to the outcome must satisfy these requirements: (1) The probability of any outcome is between 0 and 1 i.e. 0 ≤ P(Oi) ≤ 1 for each i, and (2) The sum of the probabilities of all the outcomes equals 1 i.e. P(O1) + P(O2) + … + P(Ok) = 1 6.7 P(Oi) represents the probability of outcome i
- 8. The probability of an event is the sum of the probabilities of the simple events that constitute the event. E.g. (assuming a fair die) S = {1, 2, 3, 4, 5, 6} and P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6 Then: P(EVEN) = P(2) + P(4) + P(6) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2 6.8
- 9. 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: 6.9
- 10. 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! 6.10
- 11. An event is an outcome or a set of outcomes of a random process Example: Tossing a coin three times Event A = getting exactly two heads = {HTH, HHT, THH} Example: Picking real number X between 1 and 20 Event A = chosen number is at most 8.23 = {X ≤ 8.23} Example: Tossing a fair dice Event A = result is an even number = {2, 4, 6} • Notation: P(A) = Probability of event A • Probability Rule 1: 0 ≤ P(A) ≤ 1 for any event A June 3, 2008 11Stat 111 - Lecture 6 - Probability
- 12. Events A and B are independent if knowing that A occurs does not affect the probability that B occurs Example: tossing two coins Event A = first coin is a head Event B = second coin is a head Disjoint events cannot be independent! • If A and B can not occur together (disjoint), then knowing that A occurs does change probability that B occurs Probability Rule 5: If A and B are independent P(A and B) = P(A) x P(B) June 3, 2008 12Stat 111 - Lecture 6 - Probability Independent multiplication rule for independent events
- 13. 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. Two events A and B are said to be independent if P(A|B) = P(A) and P(B|A) = P(B) P(you have a flat tire going home/radio quits working) 6.13
- 14. 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
- 17. A graduate statistics course has seven male and three female students. The professor wants to select two students at random to help her conduct a research project. What is the probability that the two students chosen are female? P(F1 * F2) = ??? Let F1 represent the event that the first student is female P(F1) = 3/10 = .30 What about the second student? P(F2 /F1) = 2/9 = .22 P(F1 * F2) = P(F1) * P(F2 /F1) = (.30)*(.22) = 0.066 NOTE: 2 events are NOT independent. 6.17
- 18. The professor in Example 6.5 is unavailable. Her replacement will teach two classes. His style is to select one student at random and pick on him or her in the class. What is the probability that the two students chosen are female? Both classes have 3 female and 7 male students. P(F1 * F2) = P(F1) * P(F2 /F1) = P(F1) * P(F2) = (3/10) * (3/10) = 9/100 = 0.09 NOTE: 2 events ARE independent. 6.18
- 19. The Addition Rule The probability that at least one of the events A or B will occur, P(A or B), is given by: If events A and B are mutually exclusive, then the addition rule is simplified to: This simplified rule can be extended to any number of mutually exclusive events. )()()()()( BAPBPAPBAPBorAP )()()()( BPAPBAPBorAP
- 20. If and A and B are mutually exclusive the occurrence of one event makes the other one impossible. This means that P(A and B) = P(A * B) = 0 The addition rule for mutually exclusive events is P(A or B) = P(A) + P(B) Only if A and B are Mutually Exclusive. 6.20
- 21. In a large city, two newspapers are published, the Sun and the Post. The circulation departments report that 22% of the city’s households have a subscription to the Sun and 35% subscribe to the Post. A survey reveals that 6% of all households subscribe to both newspapers. What proportion of the city’s households subscribe to either newspaper? That is, what is the probability of selecting a household at random that subscribes to the Sun or the Post or both? P(Sun or Post) = P(Sun) + P(Post) – P(Sun and Post) = .22 + .35 – .06 = .51 6.21
- 22. Bayes’ Law is named for Thomas Bayes, an eighteenth century mathematician. In its most basic form, if we know P(B | A), we can apply Bayes’ Law to determine P(A | B) 6.22 P(B|A) P(A|B) for example …
- 24. Conditional Probability Discrete Random Variables Continuous Random Variables Properties of Random Variables • Means of Random Variables • Variances of Random Variables June 4, 2008 24Stat 111 - Lecture 6 - Random Variables
- 25. Conditional Probability Conditional probability is the probability of an event occurring, given that another event has already occurred. Conditional probability restricts the sample space. The conditional probability of event B occurring, given that event A has occurred, is denoted by P(B|A) and is read as “probability of B, given A.” We use conditional probability when two events occurring in sequence are not independent. In other words, the fact that the first event (event A) has occurred affects the probability that the second event (event B) will occur.
- 26. A random variable is a numerical outcome of a random process or random event Example: three tosses of a coin • S = {HHH,THH,HTH,HHT,HTT,THT,TTH,TTT} • Random variable X = number of observed tails • Possible values for X = {0,1, 2, 3} • Why do we need random variables? • We use them as a model for our observed data June 4, 2008 26Stat 111 - Lecture 6 - Random Variables
- 27. A random variable is a real valued function whose domain is the sample space of a random experiment Let us consider the experiment of tossing a coin two times in succession. The sample space of the experiment is S = {HH, HT, TH, TT}.
- 28. A discrete random variable has a finite or countable number of distinct values Discrete random variables can be summarized by listing all values along with the probabilities • Called a probability distribution Example: number of members in US families June 4, 2008 28Stat 111 - Lecture 6 - Random Variables X 2 3 4 5 6 7 P(X) 0.413 0.236 0.211 0.090 0.032 0.018
- 29. Continuous random variables have a non- countable number of values Can’t list the entire probability distribution, so we use a density curve instead of a histogram Eg. Normal density curve: June 4, 2008 29Stat 111 - Lecture 6 - Random Variables
- 30. Average of all possible values of a random variable (often called expected value) Notation: don’t want to confuse random variables with our collected data variables = mean of random variable x = mean of a data variable For continuous r.v, we again need integration to calculate the mean For discrete r.v., we can calculate the mean by hand since we can list all probabilities June 4, 2008 30Stat 111 - Lecture 6 - Random Variables
- 32. Mean is the sum of all possible values, with each value weighted by its probability: μ = Σ xi*P(xi) = x1*P(x1) + … + x12*P(x12) Example: X = sum of two dice μ = 2⋅ (1/36) + 3⋅ (2/36) + 4 ⋅ (3/36) +…+12⋅ (1/36) = 252/36 = 7 June 4, 2008 32Stat 111 - Lecture 6 - Random Variables X 2 3 4 5 6 7 8 9 10 11 12 P(X) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
- 33. Spread of all possible values of a random variable around its mean Again, we don’t want to confuse random variables with our collected data variables: 2 = variance of random variable s2 = variance of a data variable For continuous r.v, again need integration to calculate the variance For discrete r.v., can calculate the variance by hand since we can list all probabilities June 4, 2008 33Stat 111 - Lecture 6 - Random Variables
- 37. Variance is the sum of the squared deviations away from the mean of all possible values, weighted by the values probability: μ = Σ(xi-μ)*P(xi) = (x1-μ)*P(x1) + … + (x12-μ)*P(x12) Example: X = sum of two dice σ2 = (2 - 7)2⋅(1/36) + (3− 7)2⋅(2/36) +…+(12 - 7)2⋅(1/36) = 210/36 = 5.83 June 4, 2008 37Stat 111 - Lecture 6 - Random Variables X 2 3 4 5 6 7 8 9 10 11 12 P(X) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
- 38. a tossed coin shows a‘head’ or ‘tail’, a manufactured item can be ‘defective’ or ‘non- defective’, the responseit is customary to call one of the outcomes a ‘success’and the other ‘not success’ Each time we toss a coin or roll a die. we call it a trial. two outcomes usually referred as ‘success’ or ‘failure’ are called Bernoulli trials. Trials of a random experiment are called Bernoulli trials, if they satisfy