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Probability and Discrete Random Variable

Probability and Discrete Random Variable. Probability. What is Probability?. When we talk about probability , we are talking about a (mathematical) measure of how likely it is for some particular thing to happen Probability deals with chance behavior

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Probability and Discrete Random Variable

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  1. Probability and Discrete Random Variable

  2. Probability

  3. What is Probability? • When we talk about probability, we are talking about a (mathematical) measure of how likely it is for some particular thing to happen • Probability deals with chance behavior • We study outcomes, or results of experiments • Each time we conduct an experiment, we may get a different result

  4. Why study Probability? • Descriptive statistics - describing and summarizing sample data, deals with data as it is • Probability - Modeling the probabilities for sample results to occur in situations in which the population is known • The combination of the two will let us do our inferential statistics techniques.

  5. Objectives • Learn the basic concepts of probability • Understand the rules of probabilities • Compute and interpret probabilities using the empirical method • Compute and interpret probabilities using the classical method • Compute the probabilities for the compound events.

  6. Sample Space & Outcomes • Some definitions • An experiment is a repeatable process where the results are uncertain • An outcome is one specific possible result • The set of all possible outcomes is the samplespace denoted by a capital letter S • Example • Experiment … roll a fair 6 sided die • One of the outcomes … roll a “4” • The sample space … roll a “1” or “2” or “3” or “4” or “5” or “6”. So, S = {1, 2, 3, 4, 5, 6} (Include all outcomes in braces {…}.)

  7. Event • More definitions • An event is a collection of possible outcomes … we will use capital letters such as E for events • Outcomes are also sometimes called simpleevents … we will use lower case letters such as e for outcomes / simple events • Example (continued) • One of the events … E = {roll an even number} • E consists of the outcomes e2 = “roll a 2”, e4 = “roll a 4”, and e6 = “roll a 6” … we’ll write that as {2, 4, 6}

  8. Example Consider an experiment of rolling a die again. • There are 6 possible outcomes, e1 = “rolling a 1” which we’ll write as just {1}, e2 = “rolling a 2” or {2}, … • The sample space is the collection of those 6 outcomes. We write S = {1, 2, 3, 4, 5, 6} • One event of interest is E = “rolling an even number”. The event is indicated by E = {2, 4, 6}

  9. Probability of an Event • If E is an event, then we write P(E) as the probability of the event E happening • These probabilities must obey certain mathematical rules

  10. Probability Rule # 1 • Rule # 1 – the probability of any event must be greater than or equal to 0 and less than or equal to 1, i.e., • It does not make sense to say that there is a -30% chance of rain • It does not make sense to say that there is a 140% chance of rain Note – probabilities can be written as decimals (0, 0.3, 1.0), or as percents (0%, 30%, 100%), or as fractions (3/10)

  11. Probability Rule # 2 • Rule #2 – the sum of the probabilities of all the outcomes must equal 1. • If we examine all possible outcomes, one of them must happen • It does not make sense to say that there are two possibilities, one occurring with probability 20% and the other with probability 50% (where did the other 30% go?)

  12. Example On the way to work Bob’s personal judgment is that he is four times more likely to get caught in a traffic jam (TJ) than have an easy commute (EC). What values should be assigned to P(TJ) and P(EC)? Solution: Given Since Which means

  13. Probability Rule (continued) • Probability models must satisfy both of these rules • There are some special types of events • If an event is impossible, then its probability must be equal to 0 (i.e. it can never happen) • If an event is a certainty, then its probability must be equal to 1 (i.e. it always happens)

  14. Unusual Events • A more sophisticated concept • An unusualevent is one that has a low probability of occurring • This is not precise … how low is “low? • Typically, probabilities of 5% or less are considered low … events with probabilities of 5% or lower are considered unusual • However, this cutoff point can vary by the context of the problem

  15. How To Compute the Probability? The probability of an event may be obtained in three different ways: • Theoretically (a classical approach) • Empirically (an experimental approach) • Subjectively

  16. Compute Probability theoretically

  17. Equally Likely Outcomes • The classical method of calculating the probability applies to situations (or by assuming the situations) where all possible outcomes have the same probability which is called equallylikelyoutcomes • Examples • Flipping a fair coin … two outcomes (heads and tails) … both equally likely • Rolling a fair die … six outcomes (1, 2, 3, 4, 5, and 6) … all equally likely • Choosing one student out of 250 in a simple random sample … 250 outcomes … all equally likely

  18. Equally Likely Outcomes • Because all the outcomes are equally likely, then each outcome occurs with probability 1/n where n is the number of outcomes • Examples • Flipping a fair coin … two outcomes (heads and tails) … each occurs with probability 1/2 • Rolling a fair die … six outcomes (1, 2, 3, 4, 5, and 6) … each occurs with probability 1/6 • Choosing one student out of 250 in a simple random sample … 250 outcomes … each occurs with probability 1/250

  19. Theoretical Probability • The general formula is Number of ways E can occur Number of possible outcomes • If we have an experiment where • There are n equally likely outcomes (i.e. N(S) = n) • The event E consists of m of them (i.e. N(E) = m) then

  20. A More Complex Example Here we consider an example of select two subjects at random instead of just one subject: Three students (Katherine (K), Michael (M), and Dana (D)) want to go to a concert but there are only two tickets available. Two of the three students are selected at random. Question 1: What is the sample space of who goes? Solution:S = {(K,M),(K,D),(M,D)} Question 2: What is the probability that Katherine goes? Solution: Because 2 students are selected at random, each outcome in the sample space has equal chance to occur. Therefore, P( Katherine goes) = 2/3.

  21. Another Example A local automobile dealer classifies purchases by number of doors and transmission type. The table below gives the number of each classification. If one customer is selected at random, find the probability that: • The selected individual purchased a car with automatic transmission 2) The selected individual purchased a 2-door car

  22. 1) 2) Solutions Apply the formula

  23. Compute Probability empirically

  24. Empirical Probability • If we do not know the probability of a certain event E, we can conduct a series of experiments to approximate it by • This is called the empirical probability or experimental probability. It becomes a good approximation for P(E) if we have a large number of trials (the law of large numbers)

  25. Example We wish to determine what proportion of students at a certain school have type A blood • We perform an experiment (a simple random sample!) with 100 students • If 29 of those students have type A blood, then we would estimate that the proportion of students at this school with type A blood is 29%

  26. Example (continued) We wish to determine what proportion of students at a certain school have type AB blood • We perform an experiment (a simple random sample!) with 100 students • If 3 of those students have type AB blood, then we would estimate that the proportion of students at this school with type AB blood is 3% • This would be an unusual event

  27. Another Example Consider an experiment in which we roll two six-sided fair dice and record the number of 3s face up. The only possible outcomes are zero 3s, one 3, or two 3s. Here are the results after 100 rolls of these two dice, and also after 1000 rolls:

  28. 0.7 0.6 0.5 Relative Frequency 0.4 0.3 0.2 0.1 0.0 0 1 2 Three’s Face Up Using a Histogram • We can express these results (from the 1000 rolls) in terms of relative frequencies and display the results using a histogram:

  29. Continuing the Experiment If we continue this experiment for several thousandmore rolls, the relative frequency for each possible outcome will settle down and approach to a constant. This is so called thelaw of large numbers.

  30. Coin-Tossing Experiment Consider tossing a fair coin. Define the event H as the occurrence of a head. What is the probability of the event H, P(H)? • Theoretical approach – If we assume that the coin is fair, then there are two equally likely outcomes in a single toss of the coin. Intuitively, P(H) = 50%. • Empirical approach – If we do not know if the coin is fair or not. We then estimate the probability by tossing the coin many times and calculating the proportion of heads occurring. To show you the effect of applying large number of tosses on the accuracy of the estimation. What we actually do here is to toss the coin 10 times each time and repeated it 20 times. The results are shown in the next slide. We cumulate the total number of tosses over trials to compute the proportion of heads. We plot the proportions over trials in a graph as shown in the following slide. We observe that the proportion of heads tends to stabilize or settle down near 0.5 (50%). So, the proportion of heads over larger number of tosses is a better estimate of the true probability P(H).

  31. Experimental results of tossing a coin 10 times on each trial

  32. Expected value = 1/2 Trial Cumulative Relative Frequency • This stabilizing effect, or long-term average value, is often referred to • as the law of large numbers.

  33. Law of Large Numbers If the number of times an experiment is repeated is increased, the ratio of the number of successful occurrences to the number of trials will tend to approach the theoretical probability of the outcome for an individual trial • Interpretation: The law of large numbers says: the larger the number of experimental trials, the closer the empirical probability is expected to be to the true probability P(A)

  34. Subjective Probability 1. Suppose the sample space elements are not equally likely and empirical probabilities cannot be used 2. Only method available for assigning probabilities may be personal judgment 3. These probability assignments are called subjective probabilities

  35. Summary • Probabilities describe the chances of events occurring … events consisting of outcomes in a sample space • Probabilities must obey certain rules such as always being greater than or equal to 0 and less then or equal to 1. • There are various ways to compute probabilities, including empirical method and classical method for experiments with equally likely outcomes.

  36. Compute Probabilities for Compound Events

  37. Venn Diagram • Venn Diagrams provide a useful way to visualize probabilities • The entire rectangle represents the sample space S • The circle represents an event E S E

  38. Example • In the Venn diagram below • The sample space is {0, 1, 2, 3, …, 9} • The event E is {0, 1, 2} • The event F is {8, 9} • The outcomes {3}, {4}, {5}, {6}, {7} are in neither event E nor event F

  39. Mutually Exclusive Events • Two events are disjoint if they do not have any outcomes in common • Another name for this is mutuallyexclusive • Two events are disjoint if it is impossible for both to happen at the same time • E and F below are disjoint

  40. Example The following table summarizes visitors to a local amusement park: One visitor from this group is selected at random: 1) Define the event A as “the visitor purchased an all-day pass” 2) Define the event B as “the visitor selected purchased a half-day pass” 3) Define the event C as “the visitor selected is female”

  41. 3) Solutions 1)The events A and B are mutually exclusive 2) The events A and C are not mutually exclusive. The intersection of A and C can be seen in the table above or in the Venn diagram below:

  42. Addition Rule for Disjoint Events • For disjoint events, the outcomes of (E or F) can be listed as the outcomes of E followed by the outcomes of F • There are no duplicates in this list • The AdditionRule for disjoint events is P(E or F) = P(E) + P(F) • Thus we can find P(E or F) if we know both P(E) and P(F)

  43. Addition Rule for More than Two Disjoint Events • This is also true for more than two disjoint events • If E, F, G, … are all disjoint (none of them have any outcomes in common), then P(E or F or G or …) = P(E) + P(F) + P(G) + … • The Venn diagram below is an example of this

  44. Example • In rolling a fair die, what is the chance of rolling a {2 or lower} or a {6} • The probability of {2 or lower} is 2/6 • The probability of {6} is 1/6 • The two events {1, 2} and {6} are disjoint • The total probability is 2/6 + 1/6 = 3/6 = 1/2

  45. Note • The addition rule only applies to events that are disjoint • If the two events are not disjoint, then this rule must be modified • Some outcomes will be double counted • The Venn diagram below illustrates how the outcomes {1} and {3} are counted both in event E and event F

  46. Example In rolling a fair die, what is the chance of rolling a {2 or lower} or an even number? • The probability of {2 or lower} is 2/6 • The probability of {2, 4, 6} is 3/6 • The two events {1, 2} and {2, 4, 6} are not disjoint • The total probability is not 2/6 + 3/6 = 5/6 • The total probability is 4/6 because the event is {1, 2, 4, 6} Note: When we say A or B, we include outcomes either in A or in B or both.

  47. General Addition Rule • For the formula P(E) + P(F), all the outcomes that are in both events are counted twice • Thus, to compute P(EorF), these outcomes must be subtracted (once) • The GeneralAdditionRule is P(EorF) = P(E) + P(F) – P(EandF) • This rule is true both for disjoint events and for not disjoint events. when E and F are disjoint, P(EandF) = 0 which leads to P(EorF) = P(E) + P(F)

  48. Example • When choosing a card at random out of a deck of 52 cards, what is the probability of choosing a queen or a heart? • E = “choosing a queen” • F = “choosing a heart” • E and F are not disjoint (it is possible to choose the queen of hearts), so we must use the General Addition Rule

  49. Solution • P(E) = P(queen) = 4/52 • P(F) = P(heart) = 13/52 • P(E and F) = P(queen of hearts) = 1/52, so

  50. Another Example A manufacturer is testing the production of a new product on two assembly lines. A random sample of parts is selected and each part is inspected for defects. The results are summarized in the table below: Suppose a part is selected at random: 1) Find the probability the part is defective 2) Find the probability the part is produced on Line 1 3) Find the probability the part is good or produced on Line 2

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