In everyday English the use of the word "probability" is not uncommon. The "probability of occurrence of an event" to a statistician is what "quantified chance of occurrence of that event" is to an ordinary person. Show Most people have some intuitive idea about probability:
Let us look at the coin-toss experiment. 4.1 Random Experiments and Sample SpacesTossing a coin or rolling a die are examples of random experiments. Whenever we talk about probability there is a random experiment behind it. We talk about probability in the context of such an experiment. Let us define it more formally.
Example 4.1.1. The following are examples of some experiments and their sample spaces.
Events We are getting ready to talk about probability. Given a sample space, we plan to talk about probability of an outcome. We may also talk about the probability of EVENTS. What is an EVENT for us? We have the following definitions:
Now we are ready to talk about probability of an outcome or an event. If we toss a coin, then one believes that the probability that the face Head will show up is 1/2. But this is about a "normal" coin. What if we toss a loaded coin? If you have a loaded coin, you may know that the probability that the face Head will show up is 1/5. But what does this mean? How and where did you learn that, for your loaded die, the probability that the face Head will show up is 1/5? When an ordinary person makes such probability statements he/she is, in fact, talking about his/her experience. Regarding your loaded coin, you have tossed your coin many times and have experienced that about once in five tosses the face Head showed up and other times the face Tail showed up. Therefore, "you know" that the probability that the face Head will show up is 1/5. Similarly, you rolled a die many times. You have experienced that about once in six throws the face 4 shows up. Therefore, "you know" that the probability that the face 4 will show up when you throw the die is 1/6. It is a different story if you are working with a loaded die because your experience would tell you something different. That was the "real life" idea of probability. In the study of the mathematics of probability, we accept this "experience" as part of the "probability model" and do the mathematics. The mathematics of probability includes the following:
4.3. Problems on Probability Exercise 4.3.l. The following table gives the blood group distribution of a certain population.
Suppose you determine the blood group of a randomly selected person from this population.
Exercise 4.3.2. A student wants to pick a school based on the past grade distribution of the school. Following is a grade distribution of last year in a school:
Suppose you note down the grade of a randomly selected student from this school.
Exercise 4.3.3. The following table gives the probability distribution of a loaded die.
4.2 The Multiplication Rule of CountingA some probability problems involve counting. So, we will spend some time of different methods of counting. The multiplication rule states that when something takes place in several stages, to find the total number of ways it can occur we multiply the number of ways each individual stage can occur. It goes as follows: Suppose a certain job (often a selection of an item) is accomplished in r stages.
Then the number of ways the original job can be accomplished is n1 n2 n3 ... nr ways. Example 4.2.1. A household would like to install a storm door. The local store offers two brand names; each brand has four different styles and three colors. Find how many choices he has in the selection. Solution The job of picking the door is done in three stages. The first stage is to pick the brand, which we can do in two ways. The second stage is to pick the style, which we can do in four ways. Then, the third stage is to pick the color, which we can do in three ways.
The whole job of picking the doorr can be done in 2x4x3=24 ways. Example 4.2.2. Refer to the example (1.4) of rolling a die twice. We want to count the number of outcomes in the sample space S. The whole experiment could be accomplished in two stages. First, the die is rolled, and the number of outcomes for this first stage is six. The second stage is to roll the die again; the second stage also has six outcomes.
The total number of outcomes in S, by the multiplication principle, is 6x6= 36. Example 4.2.3. I want to assign the 10 seats on the first row to the 163 students in the class. How many ways we can do it?
So the total number of ways this can be done is = 163 x 162 x 161 x 160 x 159 x 158 x 157 x 156 x 155 x 154 Remark. The multiplication rule of counting has wide applications. You must correctly identify whether your counting problem can be divided into several stages of simple counting problems. Confusion may arise as follows. Example 4.2.4. Suppose that we want to form a committee of 10 students out of the 163 students in this class. We have just assigned the 10 seats in the first row to the 163 students, which can be done in 163x 162 x . . . x 154 way. Could we say that the number of ways we can form a committee of 10 out of the 163 students in this class is the same? The answer is NO. While we assigned the seats, the different assignments of the seats, in the first row, to the same group of 10 students is considered as distinct. While forming a committee, the group of 10 as a whole is counted as one committee. Without going into details, the number of ways such a committee can be formed is (163 x 162 x ... x 154) / (1 x 2 x ... x 10). Ordered and unordered selectionMany counting problems that we consider essentially are like selecting r objects (or people) from a collection of n objects (or people). There are two types of such selections. In example (4.2.3), the assignment of the 10 seats is selection of 10 students where the order in which we selected 10 students did matter. However, in example (4.2.4) order in which we pick 10 to represent in the committee did not count. The selection in (4.2.3) is an ordered-selection of r "objects" from a group n "objects." But the selection in (4.2.4) is an unordered-selection of r "objects" from a group of n "objects." We have the following definitions, formulas, and notations in this context:
Problems on 4.2 The Multiplication Rule of Counting Before you attempt any problem, review the diagram.Exercise 4.2.1.
Exercise 4.2.2. Let me compute 8P5 :
Exercise 4.2.3. Let me also compute 9C4. I like to compute as follows: 9C4 = 9P4 /4!. So, 9C4 = 9P4 /4! = 3024/24 = 126. Exercise 4.2.4. How many ways can you deal a hand of 13 cards out of a deck of 52 cards? Answer = 52C13= 52!/(13! x 39!). Exercise 4.2.5. Four financial awards (of different values) will be given to the "best" four students in a class of 163. How many possible ways can these awardees be picked? Here, the order counts. The answer is 163P4. Exercise 4.2.5. How many code words of length four you can construct out of the English alphabets? This problem is not like selecting 4 from 26 letters because we can use the same letter more than once. Use the multiplication rule.
The total number of such words = 26 x 26 x 26 x 26 =264. 4.4 Probability Spaces with Equally Likely OutcomesUnlike the above problems, in some probability spaces every outcome has an equal probability. (Such is the case when you toss a "normal" coin or roll a "normal" die.) In such cases, we say that outcomes are equally likely. When outcomes are equally likely, the probability P(E) can be computed by counting the number of outcomes in E and those of S. If S has N outcomes then we have:
Remark: Outcomes are equally likely in "normal" situations. Examples are
Example 4.4.1. Suppose we roll a (fair) die three times.
Example 4.4.2. Suppose we have to form a committee of two from a group of 15 men and 19 women.
Problems on 4.4: Counting and Probability Exercise 4.4.1. Find 5! Suppose in the World Cup Soccer tournament, group A has eight teams. Now each team of group A has to play with all the other teams in the group. Find how many games will be played among the Group A teams. Exercise 4.4.3. How many ways you can deal a hand of 13 cards from a deck of 52 cards? Exercise 4.4.4. How many ways you can deal a hand of 4 Spades, 3 Hearts, 3 Diamonds, and 3 Clubs? Exercise 4.4.5. We have 13 students in a class. How many ways we can assign the four seats in the first row? Solution Exercise 4.4.6. Programming languages sometimes use hexadecimal system (also called "hex") of numbers. In this system 16 digits are used and denoted by 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F. Suppose you form a 6-digit number in hexadecimal system.
Solution Exercise 4.4.7. You are playing bridge, and you are dealt a hand of 13 cards. Exercise 4.4.8. A committee of 9 is selected at random from a group of 11 students, 17 mothers, and 13 fathers. Exercise 4.4.9. Three scholarships of unequal values will have to be awarded to a group of 35 applicants. How many was such a selection can be made? Solution Sometimes it is easier to find the probability that an event E does not occur. Then we can use this to compute the probability that E occurs.
Problems on 4.5: The Complement of an Event Answer: P(not E) = 1 - 0.9 = .01. Sometimes it is understandable that two events E and F do not influence the occurrence of each other. For example, if you roll a die twice and E is the event that the first roll will show an odd-number face and F is the event that the second roll will show 1 or 2, then it is reasonable to assume that the occurrence of E will not influence the occurrence of F. (Describe E and F in brace notation.)
Example 4.6.1. Suppose you are dealt a hand of five cards out of a shuffled deck of twenty high-cards. (Ace, King, Queen, Jack, and 10 are the high-cards.) Example 4.6.2. Suppose you roll four fair dice. Problems on 4.6: Independent Events Exercise 4.6.1. Two university employees (Mr. Park and Mr. Jones) issue tickets to illegally parked cars. Probability of the event E that Mr. Jones will notice an illegally parked car is P(E)= 0.1, and the probability of the event F that Mr. Park will notice an illegally parked car is P(F) = 0.3. Exercise 4.6.2. Suppose the two engines of a airplane function independently. Probability that the first engine fails in a flight is .01, and the probability that the second engine fails in a flight is .02. Exercise 4.6.3. The probability that you will receive a wrong number call this week is 0.3, and the probability that you will receive a sales call this week is 0.8, and the probability that you will receive a survey call this week is 0.5. What is the probability that you will receive one of each this week? (Assume independence.) Exercise 4.6.4.
The probability, of the event E,
that a student will major in either liberal arts
or in business is P(E)= .69. Find the probability
that the student will major neither in liberal arts
nor in business. In many situations the probability of an event E is described as "odds" in favor or against. This language is often used in gambling, horse races, and sports.
Example 4.7.1. Suppose you roll a die twice. Example 4.7.2. Suppose we have a game where we roll two dice. We win if the sum of the "face values" is less or equal to five; otherwise we lose.
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