Probability
3.2 Probability Measures
(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.
90 3 Probability
There are two ideas here. In every case, there is the feature of unpredictability—
a chance occurrence is one for which we cannot be certain of the outcome. The other feature is that sometimes chance is quantitative—either it can be measured exactly (the weather forecast) or else we can at least say one “chance” is greater than another (the football teams). We shall refer to these two aspects of chance as randomness and probability, respectively.
For example, say you flip a coin. There is no way to tell whether it will fall heads or tails, so we would say this is a random occurrence. If the coin is made uniformly, so that it is equally likely to show a head or a tail, we normally call it a fair coin, and we say the probability of a head is 12, and so is the probability of a tail. (Sometimes we say “50%” instead of “12”.)
We shall writeP (E)for the probability that the eventEoccurs. Suppose a fair coin is tossed, ifH means “a head is tossed” andT means “a tail is tossed”, then what we have just said is
P (H )=P (T )= 1 2.
Suppose the coin in our example was not uniform; for example, say it was made as a sandwich of disks of metal, like a quarter, but the different disks were of different densities. It might be that, if we flipped it enough times, heads would come up 90%
of the time and tails only 10%. Then we would say the probability of a head is 109. However, this still qualifies as random, because no one knows beforehand whether a particular flip will be one of the (more common) ones that results in a head or one of those that give a tail. The probabilities do not have to be equal in order for an event to be random. (However, when we say one object is selected “at random” from a collection, this normally means that every object is equally likely to be chosen.)
Probability Distributions
Given any experiment, we could list the probabilities of all the outcomes, like the list
“P (H ) =P (T )= 12” in the fair coin toss. Such a list is called the probability dis- tribution of the experiment. Of course, we don’t always know what the probabilities are, and sometimes the whole point of an investigation is to find out the probability distribution.
Sample Problem 3.7. A fair die is rolled.Si is the event that the number i is rolled fori =1,2,3,4,5,6.Emeans the roll of an odd number, andF the roll of a number less than 3. Find the probabilities of these events.
Solution. Since the die is fair,
P (S1)=P (S2)=P (S3)=P (S4)=P (S5)=P (S6)=1/6.
Clearly,P (E)=12, andP (F )= 13.
Your Turn. A single die is rolled. What are the probabilities of the following events:
E: A 4 or a 5 is rolled;
F: An even number is rolled;
G: An odd number greater than 2 is rolled?
Observe that in the above example,
P (E)=P (S1)+P (S3)+P (S5),
so thatP (E)equals the sum of the probabilities of the outcomes making upE, and similarly
P (F )=P (S1)+P (S2).
This additive property works in general. SupposeEis the event{s1, s2, . . . , sm}; as usual, we just writesi for the simple event{si}. Then
P (E)=P (s1)+P (s2)+ · · · +P (sm).
We use this to make a general definition of probability.
Definition. Consider an experiment with sample spaceS= {s1, s2, . . . , sm}. A prob- ability distribution for the experiment is a functionP with the following properties:
1. For eachsi,1≤i≤n, P (si)is a real number and 0≤P (si)≤1;
2. P (s1)+P (s2)+ · · · +P (sm)=1.
We now define the probability of the eventEby the formula P (E)=
s∈E
P (s).
Uniform Experiments
A very important case is a probability distribution in which every sample point has the same probability. In this case, the probability is called uniform. We also refer to the experiment as a uniform experiment. If a uniform experiment has a sample space withnelements, then each sample point has probability n1. The toss of a fair coin is a uniform experiment withn=2 sample points. The roll of a fair die is a uniform experiment withn =6 sample points. Another way of saying that an experiment is uniform is to call it an experiment with equally likely outcomes.
92 3 Probability
Suppose a uniform experiment has sample spaceS. then the probability of an eventEis
P (E)= |E|
|S|.
Remember: this important formula only applies to uniform experiments.
Sample Problem 3.8. A black die and a white die are thrown simultaneously.
What is the probability that the numbers shown total 8, given that the dice are fair?
Solution. Let us write(x, y)to mean thatxshows on the black die andyshows on the white die. Then there are 36 possible outcomes, namely(1,1), (1,2), . . . , (1,6), (2,1), . . . , (6,6), and they are equally likely. Five of these outcomes—
(6,2), (5,3), (4,4), (3,5), and(2,6)—give a total of 8. So P (E)=|E|
|S| = 5 36.
Your Turn. A quarter and a nickel are flipped simultaneously. What is the prob- ability that exactly one head shows, assuming that both coins are fair?
Sample Problem 3.9. A deck of cards is shuffled and one card is dealt. What is the probability that it is a spade?
Solution. There are 52 cards in a deck, of which 13 are spades. So P (E)= |E|
|S| = 13 52= 1
4.
Your Turn. A deck is shuffled and one card dealt face up. What is the probability that is a picture card (King, Queen, or Jack)?
Non-uniform Experiments
Not all experiments are uniform. But in many cases, given an experimentA, we can find a uniform experimentBsuch that the outcomes ofAare events (not necessarily simple) ofB. We shall explain this method by an example.
Consider the following experiment. Two fair dice are rolled, and the total of the points on them is recorded. This experiment has eleven outcomes which we shall denote s2, s3, . . . , s12;si is the outcome “total i”. To calculate the probability of outcomei, we shall look at a slightly different experiment. In it, two fair dice are rolled, and the result is recorded as an ordered pair of digits—(1,3)means “1 on die 1, 3 on die 2”. There are 62=36 outcomes, and each has probability 361. In this experiment, we writeEi for the event that the two numbers showing add toi. Then
E2= {11} |E2| =1 soP (E2)= 361 E3= {12,21} |E3| =2 soP (E3)= 362 = 181 E4= {13,22,31} |E4| =3 soP (E4)= 363 = 121 E5= {14,23,32,41} |E5| =4 soP (E5)= 364 = 19 E6= {15,24,33,42,51} |E6| =5 soP (E6)= 365 E7= {16,25,34,43,52,61} |E7| =6 soP (E7)= 366 = 16 E8= {26,35,44,53,62} |E8| =5 soP (E8)= 365 E9= {36,45,54,63} |E9| =4 soP (E9)= 364 = 19 E10= {46,55,64} |E10| =3 soP (E10)= 363 = 121; E11= {56,65} |E11| =2 soP (E11)= 362 = 181 E12= {66} |E12| =1 soP (E12)= 361
Table 3.1. Probabilities when rolling two fair dice
P (E)=|Ei| 36 ,
and we can calculate all the probabilities; see Table3.1. But in each case the proba- bility of the outcomesi in the first experiment is obviously equal to the probability ofEi in the second experiment. SoP (s2)= 361, P (s3)= 362, and so on.
Sample Problem 3.10. Three coins are flipped and the number of heads is recorded. What are the possible outcomes and what is the probability distribu- tion?
Solution. The outcomes are the four numbers 0, 1, 2, and 3. To calculate proba- bilities, suppose three coins were flipped and all results recorded.Eiis the event
“there areiheads showing”. Then|S| =8 and
E0= {T T T}, P (E0)= 1 8; E1= {H T T , T H T , T T H}, P (E1)= 3
8; E2= {H H T , H T H, T H H}, P (E2)= 3
8; E3= {H H H}, P (E3)= 1
8.
Your Turn. A game is played using two dice each of which has the numbers 1, 2, 3 on its faces (so each number appears twice per die). An experiment consists of rolling the dice and adding the numbers showing. What is the sample space?
What is the probability distribution?
94 3 Probability
Sample Problem 3.11. There are five marbles—two blue, three red—in a box.
One is drawn out at random. What is the probability that it is blue?
Solution. If the marbles were markedv, w, x, y, z, wherevandware blue and the others are red, then the event “blue is chosen” is{v, w}and its probability is
P (E)=|E|
|S| = 2 5.
Your Turn. There are two red, three white, and four blue marbles in a box. One is drawn at random. What is the probability that it is not blue?
Exercises 3.2 A
1. A sample space contains four outcomes:s1, s2, s3, s4. In each case, do the prob- abilities shown form a probability distribution? If not, why not?
(i) P (s1)=0.2, P (s2)=0.2, P (s3)=0.2, P (s4)=0.4;
(ii) P (s1)=0.2, P (s2)=0.3, P (s3)=0.4, P (s4)=0.5;
(iii) P (s1)=0.0, P (s2)=0.2, P (s3)=0.4, P (s4)=0.4;
(iv) P (s1)=0.5, P (s2)=0.4, P (s3)=0.3, P (s4)= −0.2.
2. A fair coin is tossed three times. What are the probabilities that:
(i) At least two heads appear;
(ii) An odd number of heads appear;
(iii) The first two results are tails?
3. In the game of roulette, a wheel is divided into 38 equal parts, labeled with the numbers from 1 to 36, 0, and 00. The spin of the wheel causes the ball to be randomly placed in one of the parts; the chances of the ball landing on any part are equal. Half of the parts numbered from 1 to 36 are red and half are black; the 0 and 00 are green. If the chances of the ball landing in any one part are equal, what are the probabilities of the following events:
(i) The ball lands on a red number;
(ii) The ball lands on a black number;
(iii) The ball lands on a green number;
(iv) The ball lands on 17;
(v) The ball lands on a number from 25 to 36 inclusive.
4. Two fair dice are rolled. What are the probabilities of the following events?
(i) The total is 9;
(ii) The total is odd;
(iii) One odd and one even number are shown.
5. A box contains seven red and five white balls. One ball is drawn at random. What is the probability that it is white?
6. In a lottery there are 90 losing tickets and 10 winning tickets. You drawn one ticket at random. What is the probability that it is a winner?
7. A random number from 0 to 99 is chosen by a computer. What are the probabil- ities of the following events?
(i) The number is even;
(ii) The number ends in 5;
(iii) The number is divisible by 11.
8. Box A contains tickets numbered 1, 2, 3, 4. Box B contains tickets numbered 2, 3, 4, 5. One ticket is selected from each box.
(i) List all the elements of the sample space of this experiment;
(ii) Find the probability that the tickets have the same number;
(iii) Find the probability that the sum of the two selected numbers is even.
9. A box contains one red, two blue, and two white marbles. One marble is selected at random. What are the probabilities that it is
(i) red? (ii) blue? (iii) white?
10. A bag contains three red marbles and five blue marbles. Three marbles are drawn simultaneously, at random, and the colors are noted.
(i) What are the outcomes of this experiment?
(ii) Calculate the probabilities of the outcomes.
Exercises 3.2 B
1. A sample space contains four outcomes:s1, s2, s3, s4. In each case, do the prob- abilities shown form a probability distribution? If not, why not?
(i) P (s1)=0.2, P (s2)=0.4, P (s3)=0.3, P (s4)=0.1;
(ii) P (s1)=0.2, P (s2)=0.4, P (s3)=0.6, P (s4)=0.2;
(iii) P (s1)=0.1, P (s2)=0.2, P (s3)=0.5, P (s4)=0.2;
(iv) P (s1)=0.1, P (s2)=0.2, P (s3)=0.3, P (s4)=1.2.
2. A fair coin is tossed four times. What are the probabilities that:
(i) At least three heads appear;
(ii) An even number of heads appear;
(iii) The first result is a head?
3. Two fair dice are rolled. What are the probabilities of the following events?
96 3 Probability (i) The total is 6;
(ii) The total is even;
(iii) Both scores are even.
4. Two fair dice are rolled. What are the probabilities of the following events?
(i) The total is 7;
(ii) The total is 8;
(iii) The total is a multiple of 3;
(iv) The total is a multiple of 4;
(v) One score is even, the other odd.
5. A box contains 12 cards, one for each month of the year. A card is drawn at random.
(i) What is the probability that the selected card is March?
(ii) What is the probability that the selected card is from a month with r in its name?
6. A box contains six red and four white balls. One ball is drawn at random. What is the probability that it is white?
7. A box contains the 365 pages from a desk calendar, one for each day of the year (not a leap year). A page is drawn at random.
(i) What is the probability that the selected page shows a day in March?
(ii) What is the probability that the selected card shows a day in summer (June 22 to September 21 inclusive)?
8. The weather forecast gives a 25% chance of snow tomorrow and a 35% chance of rain. There is a 10% chance that it will both snow and rain during the day.
(i) Represent the data in a Venn diagram.
(ii) What is the probability that it will snow but not rain?
(iii) What is the probability that it will neither snow nor rain?
9. The moose in a Canadian park are 45% plain brown, 35% mottled, and 20%
spotted. One moose is captured at random for the Bronx Zoo. What are the prob- abilities that:
(i) It is spotted?
(ii) It is not spotted?
(iii) It is not mottled?
10. There are 20 people in a class. There are 12 men (eight physics and four chem- istry majors) and eight women (five physics and three chemistry majors). One student’s name is selected at random. What is the probability that:
(i) The student is a chemistry major?
(ii) The student is male?
11. A box contains four red, two blue, and three white marbles. One is selected at random. What is the probability that:
(i) The marble is blue?
(ii) The marble is not blue?
12. A box contains six red, three blue, and five white marbles. One is selected at random. What is the probability that:
(i) The marble is blue?
(ii) The marble is not white?
13. An examination has two questions. Of 100 students 75 do Question 1 correctly and 72 do Question 2 correctly. 64 do both questions correctly.
(i) Represent these data in a Venn diagram.
(ii) A student’s answer book is chosen at random. What is the probability that:
(a) Question 1 contains an error?
(b) Exactly one question contains an error?
(c) At least one question contains an error?
14. There are 15 members in your club. There are seven men and eight women.
The club committee has three members. One member’s name is selected at ran- dom.
(i) What is the probability that:
(a) The person selected is on the committee?
(b) The person selected is male?
(ii) Can you work out the probability that the person selected is a male commit- tee member?