Probability
3.1 Events
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Fig. 3.1. Tree diagrams
Solution. {1,2}.
Your Turn. In the above game, what is the event “Betty wins”?
Tree Diagrams
Consider the experiment where a coin is tossed; the possible outcomes are “heads”
and “tails” (HandT). The possible outcomes can be shown in a diagram like the one in Figure3.1(a). A special point (called a vertex) is drawn to represent the start of the experiment, and lines are drawn from it to further vertices representing the outcomes, which are called the first generation. If the experiment has two or more stages, the second stage is drawn onto the outcome vertex of the first stage, and all these form the second generation, and so on. The result is called a tree diagram. Figure3.1(a) shows the tree diagram for the experiment where the coin is tossed twice.
Sample Problem 3.2.
(i) A coin is flipped three times; each time the result is written down. What is the sample space? Draw a tree diagram for the experiment.
(ii) Repeat this example in the case where you stop flipping as soon as a head is obtained.
Solution. (i) The sample space is{H H H, H H T , H T H, H T T , T H H, T H T , T T H, T T T}. The tree diagram is shown in Figure3.2(a).
(ii) The sample space is{H, T H, T T H, T T T}. The tree diagram is shown in Figure3.2(b).
Sample Problem 3.3. An experiment consists of flipping three identical coins si- multaneously and recording the results. What is the sample space?
Solution. {H H H, H H T , H T T , T T T}. Notice that the order is not relevant here, so that for example, the eventsH H T,H T H andT H H of Sample Prob- lem3.2are all the same event in this case.
Fig. 3.2. Tree diagrams for Sample Problem3.2
Your Turn. An experiment consists of flipping a quarter and noting the result, then flipping two pennies and noting the number of heads. What is the sample space? Draw a tree diagram.
One common application of tree diagrams is the family tree, which records the descendants of an individual. The start is the person, the first generation represents his or her children, the second the children’s children, and so on. (This is why the word “generation” is used.)
Set Language
Since events are sets, we can use the language of set theory in describing them.
We define the union and intersection of two events to be the events whose sets are the union and intersection of the sets corresponding to the two events. For exam- ple, in Sample Problem3.1, the event “either John wins or the roll is odd” is the set {1,2} ∪ {1,3,5} = {1,2,3,5}. (For events, just as for sets, “or” carries the under- stood meaning, “or both”.) The complement of an event is defined to have associated with it the complement of the original set in the sample space; the usual interpreta- tion of the complementEof the eventEis “Edoesn’t occur”. Venn diagrams can represent events, just as they can represent sets.
The language of events is different from the language of sets in a few cases. IfS is the sample space, then the eventsSand∅are called “certain” and “impossible”. If UandV have empty intersection, they are disjoint sets, but we call them mutually exclusive events.
Sample Problem 3.4. A die is thrown. Represent the following events in a Venn diagram:
A: An odd number is thrown;
B: A number less than 5 is thrown;
C: A number divisible by 3 is thrown.
84 3 Probability Solution.
Your Turn. Repeat the above example for the events:
A: An even number is thrown;
B: John wins (in the game of Sample Problem3.1);
C: Betty wins.
Sample Problem 3.5. A coin is tossed three times. EventsA,B, andC are de- fined as follows:
A: The number of heads is even;
B: The number of tails is even;
C: The first toss comes down heads.
Write down the outcomes inA,B,C,A∪C,B∩C, andC. Are any of the sets mutually exclusive? Write a brief description, in words, ofB∩C.
Solution. A = {H H T, H T H,T H H, T T T}, B = {H H H,H T T,T H T, T T H},C = {H H H, H H T , H T H, H T T},A ∪ C = {H H H,H H T,H T H, H T T,T H H,T T T},B∩C= {H H H,H T T}, andC= {T H H, T H T , T T H, T T T}.AandBare mutually exclusive, as areAandB∩C.B∩C consists of all outcomes with a head first and the other two results equal.
Your Turn. For the sets defined above, write down the outcomes inA ∩ C, B ∪ C, andA.
Sample Problem 3.6. Describe an experiment in which the outcomes corre- spond to the time one must wait in the checkout line at a supermarket.
Solution. The experiment might consist of watching the next person to come to the checkout in a supermarket and observing the time, in minutes, until he or she is served. The outcome can be any non-negative real number (although large numbers are extremely unlikely). This example shows that sample spaces need not be finite sets.
Exercises 3.1 A
1. A die is rolled and a coin is tossed, and the results are recorded.
(i) Write down all members of the sample space.
(ii) Write down all members of the event E: The die roll is even.
2. An experiment consists of studying families with three children. B represents
“boy”,Grepresents “girl”, and, for example,BGGwill represent a family where the oldest child is a boy and the young children are both girls.
(i) What is the sample space for this experiment?
(ii) We define the following events:
E: The oldest child is a boy;
F: There are exactly two boys.
(a) What are the members ofEandF? (b) Describe in words the eventE∩F.
(iii) Draw a Venn diagram for this experiment, and show the eventsE andF on it.
3. Electronic components are being inspected. Initially one is selected from a batch and tested. If it fails, the batch is rejected. Otherwise a second is selected and tested. If the second fails, the batch is rejected; otherwise a third is selected. If all three pass the test, the batch is accepted.
(i) Draw a tree diagram for this experiment.
(ii) How many outcomes are there in the sample space?
(iii) What is the event, “fewer than three components are tested”?
4. An experimenter tosses four coins—two quarters and two nickels—and records the number of heads. For example, two heads on the quarters and one on the nickels is recorded(2,1).
(i) Write down all members of the sample space.
(ii) Write down all members of the events:
E: There are more heads on the quarters than on the nickels.
F: There are exactly two heads in total.
G: The number of heads is even.
(iii) Write down all members of the eventsE∪F,E∩F,E∩G,F ∩G.
(iv) Describe in words the eventsE∩F andF ∩G.
5. A coin is tossed. If the first toss is a head, the experiment stops. Otherwise, the coin is tossed a second time.
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(i) Draw a tree diagram for this experiment.
(ii) List the members of the sample space.
(iii) What is the event, “the coin is tossed twice”?
6. A student takes courses in mathematics and computer science. Define the events EandBby:
E: She passes mathematics;
F: She passes computer science.
Write symbolic expressions (usingE, F,∩,∪and – ) for the following events:
(i) The student fails mathematics;
(ii) The student passes both subjects;
(iii) The student passes exactly one subject;
(iv) The student passes at least one subject;
(v) The student fails both subjects;
(vi) The student passes mathematics but fails computer science.
7. In each of the following cases, are the two events mutually exclusive?
(i) A die shows a 4;
A die shows an odd number.
(ii) A die shows a 4;
A die shows a perfect square.
(iii) A person is male;
The same person is a store clerk.
(iv) A person is a college freshman;
The same person is a college sophomore.
8. SupposeEandF are two events. MustEand(E∪F )be mutually exclusive?
Exercises 3.1 B
1. A die is rolled twice, and the results are recorded as an ordered pair.
(i) How many outcomes are there in the sample space?
(ii) Consider the events:
E: The sum of the throws is 4;
F: Both throws are even;
G: The first throw was 3.
(a) List the members of eventsE, F, G, E∪F, E∩F, E∩G, F∩G.
(b) Write descriptions in words of the eventsE∩F, F∩G.
(c) Are any two of the eventsE, F, Gmutually exclusive?
(iii) Draw a tree diagram for this experiment.
(iv) Represent the outcomes of this experiment in a Venn diagram. Show the eventsE,F, andGin the diagram.
2. A bag contains two red, two yellow, and three blue balls. In an experiment, one ball is drawn from the bag and its color is noted, and then a second ball is drawn and its color noted.
(i) Draw a tree diagram for this experiment.
(ii) How many outcomes are there in the sample space?
(iii) What is the event, “two balls of the same color are selected”?
3. In an experiment, video tapes are tested until either two defectives have been found or four tapes have been tested in total.
(i) How many outcomes are there in the sample space for this experiment?
(ii) What is the event, “fewer than four tapes are tested”?
4. Two dice, one red and one green, are rolled and the results recorded: for example, 53 means “5 on red, 3 on green”.
(i) Write down all members of the sample space.
(ii) Write down all members of the events:
E: The total is 4;
F: The total is 11;
G: There is at least one 6.
(iii) Write down all members of the eventsE∪F,E∩F,F∩G,F ∩G.
(iv) Describe in words the eventsF ∩GandF ∩G.
5. Repeat the preceding Exercise for the case where the two dice are the same color and indistinguishable (the notation 53 will mean “5 on one die, 3 on the other”).
6. Your doctor tests your cholesterol level each month. After the initial reading she makes three tests, and records whether the reading is higher than (H), the same as (S) or lower than (L) the preceding month.
(i) What are the possible outcomes of this experiment? Write down all mem- bers of the sample space.
(ii) Consider the events:
E: The cholesterol level never decreases;
F: The cholesterol level decreases at least twice.
(a) Write down the elements ofE,F andF. (b) Write a description in words of the setsE, F. (c) AreEandF mutually exclusive?
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7. Workers in a factory are classified by gender, experience and union member- ship.
(i) Define events as follows:
E: The worker is male;
F: The worker has one year’s experience at least;
G: The worker belongs to the union.
Write descriptions in words of the following events:E, F,G,E∩G,E∩E, E∪E,G∩F,E∩F,E∩F ∩G.
(ii) Write expressions in symbols (usingE, F, G,∩,∪and –) for the events:
(a) The worker is a woman who belongs to the union;
(b) The worker has less than one year’s experience and is a man;
(c) The worker has at least one year’s experience and is a woman union member.
(iii) Represent the set of all workers on a Venn diagram, with the setsE,F and Gshown. Shade in the set defined in (ii)(c) above.
8. An experiment consists of drawing three marbles from a jar, one at a time.
Brepresents “blue”,Grepresents “green”,Rrepresents “red”, and, for example, BGGwill represent a drawing where the first marble was blue and the second and third were both green.
(i) What is the sample space for this experiment?
(ii) We define the following events:
E: The first marble is green;
F: There are exactly two red marbles.
(a) What are the members ofEandF? (b) Describe in words the eventE∩F. (c) AreEandF mutually exclusive?
9. Consider the following pairs of events. Which pairs are mutually exclusive?
(i) A coin is tossed and falls heads.
The coin falls tails.
(ii) A student wears a watch.
The student wears sneakers.
(iii) Joe is a freshman.
Joe is a sophomore.
(iv) A die is rolled and gives an odd number.
The roll is greater than 4.
(v) Two dice show a total of 4.
One of the dice shows a 2.
(vi) Two dice show a total of 4.
One of the dice shows a 6.
(vii) A person is male.
The same person is female.
(viii) A student is enrolled in a Physics course.
The same student is enrolled in a Psychology course.
10. The eventsEandF are mutually exclusive. Which of the following are mutually exclusive pairs? Draw a Venn diagram showing the intersection in each case:
(i) EandF; (ii) EandE; (iii) EandF. 11. EandF are two events.
(i) Write symbolic expressions (using the symbolsE, F,∩,∪and−) for the events:
(a) Eoccurs butF does not;
(b) EandF both occur;
(c) Edoes not occur;
(d) Exactly one of the two occurs.
(ii) Illustrate the above events in Venn diagrams.
12. E, F, Gare three sets.
(i) Write symbolic expressions (using the symbolsE, F, G,∩,∪and−) for the events:
(a) Eoccurs but neitherF norGdoes;
(b) EandF both occur (no information is given aboutG);
(c) Edoes not occur; at least one ofF andGdoes;
(d) Precisely two of the three occur.
(ii) Illustrate the above events in Venn diagrams.
13. Draw tree diagrams for your family, starting with your grandparents.