An example to consider Say I take a quarter and one of the two dice (what is known as one die) that I have from old board games and I shake the quarter and the die in my hands and throw them on the ground. I might be interested in which side of the coin is face up. Similarly I might be interested in which number is face up on the die. The most basic outcomes on the coin are heads and tails. The most basic outcomes on the die are 1, 2, 3, 4, 5, and 6.
Example continued Let’s say that the result of the coin flip is one variable with values head or tails. Another variable is the result of the die roll. Each possible outcome of a variable is referred to as an event. A simple event is described by a single characteristic. So, in my tossing a coin and rolling a die example a simple event is what happens with only one of the items: the coin or the die. Events may be more complex than the possible outcomes of a variable.
Example continued A joint event is an event that has two or more characteristics. From our example a joint event might be the coin lands heads up and the die lands 4 up. The complement of event A, written A’, includes all events that are not part of A. The complement of heads is tails. The collection of all possible events is the sample space.
Types of Probability There are three types of probabilities: a priori probability, empirical probability (the relative frequency method) and subjective probability. A priori probability (where the phrase is italicized and you start out by saying the same thing you say when a doctor puts a popsicle stick in your mouth and then you say pre or e altogether) is a situation where probability is assigned based on prior knowledge of the process involved. Examples of this are that we can assign probability in card games, coin flipping, and die tossing. Other examples will be seen in later chapters.
Types of Probability The empirical approach to assigning probability is used when data is available about the past history of the experiment. The probability of an outcome is the relative frequency of the outcome. You can form this probability by taking the ratio of the number of times the outcome came up over the total number of times of the experiment. As an example, if out of 100 sales calls, you had 37 sales, the probability of a sale would be 37/100 = .37. The subjective approach to assigning probability is used when the other two can not be used. You and a friend may consider the experiment that Nebraska will win the national championship in football this year. The outcomes are either they will win or they will not win. You may say P(win) = .4 (meaning the probability of a win is .4) Thus you also say P(not win) = .6. Your friend might say the probabilities should be .7, .3, respectively. Your differences represent your subjectivity.
Business Topics So, in my example here I had a coin flip and a roll of the die. But this is a class in business. We might talk about did a sale occur or not. Maybe we will be interested in the amount of sales. The universe of business examples is large. We will typically talk about one or two or three variables and each variable has basic outcomes. Remember the basic outcomes are events. But events can be more complex. As an example of a more complex event we might be interested in the event when a die is rolled the result is an even number. Or maybe when we look at store sales on a day (like at each Wal-Mart) we are interested in the outcome of sales being more than $1,000,000.
Contingency Table A contingency table is a particular way to view a sample space. An example in the book mentions that 1000 households where contacted at one point and time and each was asked if in the coming year they planned to purchase a big-screen TV. At that point they could answer yes or no. Then a year later each household was called back and asked it they made a purchase and they could say yes or no. So, here we have two variables. In a contingency table one variable makes up the rows and the other makes up the columns. The next screen has the info from the example in the book.
Contingency Table Actually Purchased Planned to purchase Yes No Total Yes 200 50 250 No 100 650 750 Total 300 700 1000 So, to create this table detailed records had to be kept for the year by the researcher(s). Note 100 people didn’t plan to make a purchase but they actually did by the end of the year. (Maybe they changed their plans after the researcher called the first time.)
Venn Diagram A Venn diagram, named in honor of Mr. Venn, is another way to present the sample space for two variables. The rectangle here represents the sample space. On one variable we have event A and that takes up the space represented by circle A. Ignoring circle B, all the rest of the rectangle is A’ (the complement of A). A similar interpretation holds for B. B A
Notation On the variable planned to purchase the TV in the coming year we saw the responses could be yes or no. Usually we use some notation for each response. For example, we might use A to mean yes, there was a plan to purchase and the A’ means there is no plan to purchase. (You could use any letter for this.) Similarly, B could mean actually purchased and B’ could mean didn’t purchase.
Simple Probability Simple probability refers to the probability of simple events. From our example the probability of simple events would be denoted P(A), P(A’), P(B) and P(B’). Now, P(A) = number who planned to purchase divided by the total number of households surveyed = 250/1000 = 0.25. This means there is a a 0.25 chance that a household planned to purchase. Note the 250 came from the contingency table as a row total. Similarly, other simple probabilities here are P(A’) = 0.75, P(B) = 0.30, and P(B’) = 0.70.
Joint Probability Actually Purchased Planned to purchase Yes No Total Yes 200 50 250 No 100 650 750 Total 300 700 1000 Here I have put the contingency table again from a few slides back. If you divide each number in the table by the grand total (here 1000 and this represents the total number of folks surveyed) the table is then called a joint probability table. Let’s do this and see what results on the next screen.
Joint Probability Actually Purchased Planned to purchase Yes No Total Yes 0.20 0.05 0.25 No 0.10 0.65 0.75 Total 0.30 0.70 1.0 Notice here that the 0.25 was what we had before as the P(A). So we have the simple probabilities in the Total column and row. In this context we call the simple probabilities marginal probabilities because they occur in the “margins” of the table. So, I will reproduce this table again with the general probability notation for the simple events.
Joint Probability Actually Purchased Planned to purchase Yes No Total Yes 0.20 0.05 P(A) No 0.10 0.65 P(A’) Total P(B) P(B’) 1.0 What are the other numbers in the table (except for the 1 in the total)? These other numbers are called joint probabilities! We often call each number an intersection of events, or we connect the two events with the word “and.” For example, P(A and B) = P(A ⋂ B) = 0.20. In words the joint probability here would be the probability that a person planned to purchase and actually made a purchase. On the next screen we have the whole table in general notation.
Joint Probability Actually Purchased Planned to purchase Yes No Total Yes P(A and B) P(A and B’) P(A) No P(A’ and B) P(A’ and B’) P(A’) Total P(B) P(B’) 1.0 For each variable here we only let the people surveyed say yes or no on each question. On the planned to purchase variable, the yes and no are mutually exclusive. This means yes and no can not occur together for one person. Plus the yes and no are the only responses permitted so together they are collectively exhaustive responses (or events). The same holds for the actually purchased variable.
Marginal Probability Note in the joint probability table on the previous screen (which is a contingency table that has been modified by dividing all numbers by the grand total!) that 1) In any row the marginal probability is the sum of the joint probabilities in that row, and 2) In any column the marginal probability is the sum of the joint probabilities in that column. (Also note that the sum of the probability of complements equals 1 – for example P(A) + P(A’) = 1
Union of Events – the General Addition Rule Sometimes we want to ask a question about the probability of A or B, written P(A or B) = P(A ⋃ B). By the general addition rule P(A or B) = P(A) + P(B) – P(A and B). In our example we have P(A or B) = 0.25 + 0.30 – 0.20 = 0.35. Let’s think about this some more. How many planned to purchase? 250! How many actually purchased? 300! But 200 of these folks were in both A and B. So, when we ask a question about A or B we want to include all that said A or B, but we only want to include them once. If they said both we subtract out the intersection.
An example Income US tax code < 50,000 > 50,000 total Fair 225 180 405 Unfair 280 320 600 Total 505 500 1005
a) All the simple events are A = fair tax code, A’= unfair tax code, B = less than $50,000 income, and B’ = more than $50,000 in income. Note, I defined A and A’ as tax code information. b) All the joint events are (using abbreviations) fair and less, fair and more, unfair and less, and unfair and more. c) The complement of a fair tax code is unfair tax code. d) Fair tax code and less than $50,000 is a joint event because it has two or more (here just two) characteristics.
Let’s convert contingency table into joint probability table (here without labels): 225/1005 180/1005 405/1005 280/1005 320/1005 600/1005 505/1005 500/1005 1005/1005 or .22 .18 .40 .29 .32 .60 .50 .50 1 where I rounded to 2 decimals and some additions are off due to this rounding (.22 + .29 = .50)
a) P(A’) = .60 b) P(A’ and B) = .29 c) P(A’ or B) = .6 + .5 - .29 = .81 d) The difference is .81 - .29 = .52 NO NO NO this is not what it means here. In the question I want you to be able to state the difference between an intersection of events (A and B) and a union of events (A or B). The intersection is usually smaller because you have to meet both events, whereas with a union you just have to satisfy only one of the events (although some will meet both).