# probability and discrete random variable. probability

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• Slide 1
• Probability and Discrete Random Variable
• Slide 2
• Probability
• Slide 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
• Slide 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.
• Slide 5
• Objectives 1.Learn the basic concepts of probability 2.Understand the rules of probabilities 3.Compute and interpret probabilities using the empirical method 4.Compute and interpret probabilities using the classical method 5.Compute the probabilities for the compound events.
• Slide 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 sample space 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 {}.)
• Slide 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 simple events 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 e 2 = roll a 2, e 4 = roll a 4, and e 6 = roll a 6 well write that as {2, 4, 6}
• Slide 8
• Example Consider an experiment of rolling a die again. There are 6 possible outcomes, e 1 = rolling a 1 which well write as just {1}, e 2 = 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}
• Slide 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
• Slide 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)
• Slide 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?)
• Slide 12
• Example On the way to work Bobs 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
• Slide 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)
• Slide 14
• Unusual Events A more sophisticated concept An unusual event 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
• Slide 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
• Slide 16
• Compute Probability theoretically
• Slide 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 equally likely outcomes 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
• Slide 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
• Slide 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
• Slide 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.
• Slide 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: 1)The selected individual purchased a car with automatic transmission 2)The selected individual purchased a 2-door car
• Slide 22
• Solutions 1) 2) Apply the formula
• Slide 23
• Compute Probability empirically
• Slide 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)
• Slide 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%
• Slide 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
• Slide 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:
• Slide 28
• Using a Histogram We can express these results (from the 1000 rolls) in terms of relative frequencies and display the results using a histogram: 012 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Relative Frequency Threes Face Up
• Slide 29
• Continuing the Experiment If we continue this experiment for several thousand more rolls, the relative frequency for each possible outcome will settle down and approach to a constant. This is so called the law of large numbers.
• Slide 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