Probability
Examples/Activities
Math B30
Probability
Examples/Activities
1. a) Given the sets A = { 2,3,4,5,6,7} and B = {1,3,5,7,9}, list the elements these sets have in common.
E.g.: what is 
?
b) For these sets A and B, list all the elements that are present. E.g.: what is 
?
c) Illustrate both and by drawing a Venn diagram.
2. Given the following Venn diagram, identify the numbers represented by each of the following:
3. Draw a Venn diagram to illustrate the following situation, and then solve the problems.
Ms. Jones is the Vice-Principal of a comprehensive high school in Saskatchewan. Part of her job is to determine the timetable for the school year. From the Grade 12 student request forms this year, she has found that 163 students requested Mathematics, 95 requested Physics, and 124 requested Chemistry, while 17 students requested none of these. As well, she noted that 61 requested Mathematics and Physics, 67 Mathematics and Chemistry, 49 Chemistry and Physics, and 27 requested all three. How many requested only Mathematics? Only Physics? Only Chemistry? How many request forms (students) were there?
1. What is the probability that a card drawn at random from a standard deck of 52 playing cards will be a king or a queen?
2. If a die is rolled, what is the probability it is a 5 or a number less than 3?
3. When rolling two dice at the same time, what is the probability of obtaining a total of 7 or a total of 11 on one roll?
4. Statistics indicate that 64% of our population lives in cities, 27% in suburban areas, and 9% in rural areas. If we are to randomly select a person to take part in a poll, what is the probability that person will be from a rural or suburban area?
5. If you toss a coin and roll a fair die, what is the probability of tossing a head or rolling a 3?
6. What is the probability of drawing a face card (jack, queen, king) or a club?
1. A bag contains 3 red marbles, 4 green marbles, and 7 blue marbles. Draw two marbles, one at a time, placing the first marble back in the bag before the second draw. What is the probability that we draw:
a) a red, then a green;
b) a red and a green;
c) a red and a blue;
d) a green and a blue;
e) two red;
f) two green;
g) two blue?
In three draws, always replacing the first marble, what is the probability of one red, one green, and one blue?
2. The probability that Jennifer will be chosen to run for her school's relay team is 2/3, while the probability that Ila will be chosen is 1/2, and for Kendra 2/5. Given that these events are independent, what is the probability that:
a) Ila and Jennifer are chosen;
b) Kendra and Jennifer are chosen;
c) Ila and Kendra are chosen;
d) Ila, Kendra, and Jennifer are chosen;
e) Jennifer and Kendra are chosen, but Ila is not;
f) Ila and Kendra are chosen, but Jennifer is not;
g) Jennifer is chosen, but Ila and Kendra are not;
h) none of them are chosen;
i) at least one of the three is chosen;
j) at least two of the three are chosen?
1. In their annual door prize draw, the employees of a store have one ticket entered for each year they have been with the company. Reid has been with the company for five years, Janet for 15 years, and Bill for 22 years. There are 258 tickets altogether. If winning tickets are discarded after being drawn, what are Reid's chances of winning the first two draws? Janet's chances of winning the first two draws? Bill's chances?
2. In drawing two cards from a standard deck of playing cards, one after the other, without replacement, what is the probability that:
a) two clubs are drawn;
b) two kings are drawn;
c) an ace and a queen are drawn;
d) two red cards are drawn; or,
e) two face cards (jacks, queens, or kings) are drawn?
3. From a drawer containing six black socks, 8 white socks, and 4 red socks, Jon reaches in and selects two without looking. What is the probability that:
a) he selects a pair of black socks;
b) he selects a pair of red socks;
c) he selects a pair of white socks; or,
d) he selects a mismatched pair?
Decide whether each of the following represents events which are mutually exclusive, independent, or dependent, and then complete the calculations to determine the indicated result.
1. What is the probability of being dealt two successive aces, if the first card is not replaced?
2. When rolling a die and tossing a coin, what is the probability of rolling a number greater than 4 or tossing a head?
3. What is the probability of drawing two hearts in succession, if the first card is replaced before the second is drawn?
4. What is the probability of rolling a 4 or a 3 on one roll of a die?
5. A student determines the probability of being accepted to the college of law is 3/5, to the college of engineering 1/2, and to the college of education 3/4. What is the probability that the student is:
a) accepted by both law and engineering;
b) accepted by engineering or education;
c) accepted by all three;
d) not accepted by any; or,
e) accepted by law or engineering or education?
Several types of activities can be suggested to generate the coefficients of a binomial expansion. One is the problem of determining how many routes there are from A to B in the following grid, if no `doubling back' is allowed.
A second activity is to request students to design an elementary `pinball' machine. In this machine, the ball is let in at the top, and must work its way to the bottom, without rebounding upwards. The students are to determine how many different paths the ball might take to reach the bottom. Two different pinball machines are drawn below.
1. Find the coefficient of the third term of (x + y)6.
2. Find the coefficient of the middle term of (b + g)8.
1. Expand (x - y)6 using the Binomial Theorem.
2. Expand (3x + y)4 using the Binomial Theorem.
3. Illustrate how many different ways a coin tossed seven times could land using the expansion of (h + t)7. How many ways can we have less than three heads?
Consider the special case where h and t are both equal to one.
4. Expand (4x - 2yz)5.
5. Set up, but do not simplify, the expansion of (2x - 5y)9.
1. In a family of five children, in how many ways might one have three girls and two boys? What is the probability of having a family of three girls and two boys? Four girls and a boy? five boys? (Note that for our purposes, we are assuming that there is an equal chance of a boy or a girl being born, whereas in the real world there is, in Canada, a slightly higher than 50% occurrence of girls being born.)
2. If Asha's success rate in completing free throws is 50%, what is the probability that she will be successful on all four free throws she is awarded in a game?
3. In a series of tossing a coin, what is the probability that a fair coin will land 'heads' 8 times in a row ?
4. If a coin is tossed 12 times, in how many ways could four heads and eight tails appear? What is the probability of this occurring? In how many ways could six of each appear? What is the probability of this occurring?
5. In a family of seven children, what is the probability of having at least five girls? at least six girls? all but one boy?
6. What is the seventh term of the expansion of (4x3 - 2y)9?
7. In a true-false test of twelve questions, how many different ways can five true and seven false answers be arranged? How many different test solutions are possible?