MDM4U Unit 2 Master
Overview of Unit 2: (Chapter # 5 – Nelson)
Lesson 1: / Sec.5.1 Probability Distributions & Expected ValueLesson 2: / Sec.5.2a Pascal’s Triangle & Properties
Lesson 3: / Sec.5.2b Binomial Theorem Day # 1
Lesson 4: / Sec.5.2b Binomial Theorem Day # 2
Lesson 5: / Sec.5.3 Binomial Distributions
Lesson 6: / Sec.5.4 Normal Approximation to the Binomial Distribution
Lesson 7: / Geometric Distributions
Lesson 8: / Hypergeometric Distributions
Lesson 9: / Unit Review Day # 1
Lesson 10: / Unit Review Day # 2
Unit 2: Homework
Topic / HomeworkProbability Distributions & Expected Value / p.277 # 4 , 7 – 9 , 15 , 16 , 18
Pascal’s Triangle / YES
Binomial Theorem Day # 1 / p. 289 # 2 , 5 , 13abc
Binomial Theorem Day # 2 / p. 289 # 3 , 4 , 6 , 7 , 8 , 11 , 12 , 13d, 14 , 15abc , 20 , 21 , 23
Binomial Distributions / p. 299 # 1ab part i and ii , 6abc , 7abc, 9abc , 10ab part i
Normal Approximation to the Binomial Distribution / p.311 # 5 , 7 , 8 , 10 , 12
Geometric Distributions / YES
Hypergeometric Distributions / YES
Pascal’s Triangle & Properties Homework
1. In the following arrangements of letters, start from the top and proceed to the next row by moving diagonally left or diagonally right. How many different paths will spell out each word?
a) P
A A
T T T
T T T T
E E E E E
R R R R R R
N N N N N N N
S S S S S S S S
b) M
A A
T T T
H H H H
E E E E E
M M M M M M
A A A A A A A
T T T T T T
I I I I I
C C C C
S S S
2. Express as a single term from Pascal’s Triangle.
a) t 7 , 2 + t 7 , 3 b) t 51 , 40 + t 51 , 41 c) t 18 , 12 - t 17 , 12 d) t n ,r - t n-1 , r
3. Determine the sum of the terms in each of these rows in Pascal’s triangle.
a) row 12 b) row 20
4. Determine the row number for each of the following row sums from Pascal’s triangle.
a) 256 b) 2048 c) 16 384 d) 65 536
5. Determine the number of possible routes from A to B if you can only travel South or East. Assume North is the top of the page.
a) A
B
b) A
B
6. A checker is placed on a checkerboard as shown bellow. The checker may move diagonally upward. Although it cannot move into a square with an X, the checker may jump over the X into the diagonally opposite square.
XX
●
a) How many paths are there to the top of the board?
b) How many paths would there be if the checker could move both diagonally and straight upward?
Geometric Distributions Homework
1. A teacher provides pizza for his class if they earn an A-average on any test. The probability of the class
getting an A-Average on one of his tests is 8%.
a) What is the probability that the class will earn a pizza on the fifth test?
0.0573
b) What is the probability that the class will not earn a pizza for the first seven tests?
0.0446
c) What is the expected waiting time before the class gets a pizza?
11.5 tests
2. A poll indicated that 34% of the population agreed with a recent policy paper issued by the government.
a) What is the probability that the pollster would have to interview five people before finding a
supporter of the policy?
0.0426
b) What is the expected waiting time before the pollster interviews someone who agrees with the policy?
1.9 people
3. Suppose that 1 out of 50 cards in a scratch-and-win promotion gives a prize.
a) What is the probability of winning on your fourth try?
0.0188
b) What is the probability of winning within your first four tries?
0.0776
c) What is the expected number of cards you would have to try before winning?
49 cards
4. A top NHL hockey player scores on 93% of his shots in a shooting competition.
a) What is the probability that the player will not miss the goal until his 20th try?
0.0176
b) What is the expected number of shots before he misses?
13.3 shots
Hypergeometric Distributions Homework
1. In a mathematics class of 20 students, 5 are bilingual. If the class is randomly divided into 4 project
teams,
a) What is the probability that a team has fewer than 2 bilingual students?
0.6339
b) What is the expected number of bilingual students on a team?
1.25
2. In a swim meet, there are 16 competitors, 5 of whom are from the Eastern Swim Club.
a) What is the probability that 2 of the 5 swimmers in the first heat are from the Eastern Swim Club?
0.3777
b) What is the expected number of Eastern Swim Club members in the first heat?
1.6
3. The door prizes at a dance are four $10 gift certificates, five $20 gift certificates, and three $50 gift
certificates. The prize envelopes are mixed together in a bag, and five prizes are drawn at random.
a) What is the probability that none of the prizes is a $10 gift certificate?
0.0707
b) What is the expected number of $20 gift certificates drawn?
2.1
4. A 12-member jury for a criminal case will be selected from a pool of 14 men and 11 women.
a) What is the probability that the jury will have 6 men and 6 women?
0.2668
b) What is the probability that at least 3 jurors will be women?
0.9886
c) What is the expected number of women on the jury?
5.3
5. Seven cards are dealt from a standard deck.
a) What is the probability that three of the seven cards are hearts?
0.1758
b) What is the expected number of hearts?
1.8
Probability Distribution& Expected Value
To define probability distribution, the probability of each outcome must be determined. A graph or a table may be used to present the probabilities of each outcome.
Example # 1:
Determine the probability distribution for the sum of the roll of two dice.
Solution #1:
SUM / PROBABILITY2 /
3 /
4 /
5 /
6 /
7 /
8 /
9 /
10 /
11 /
12 /
To construct a probability model it is necessary to assign a numerical value to each outcome. The assigned value to a real life occurrence is called the random variable and is denoted by the letter X.
In Example # 1, the random variable X was the sum of the two dice. The definition of a random variable will depend on the experiment.
Example # 2:
Define what the appropriate random variable is for the experiments and indicate all possible values in set notation.
a) A machine produces bolts and the manufacturer is interested in the number of bolts produced whose shafts are longer than 60 cm.
b) A task force committee is to be selected and it is important that there is a fair representation of males on the committee.
Solution #2a:
a) Let X be the number of bolts with length longer than 60 cm.
X = { 0 , 1 , 2 , 3 , ….. }
Solution #2b:
b) Let X be the number of males in the committee.
X = { 0 , 1 , 2 }
Expected Value:
The predicted average of all possible outcomes of a probability experiment. The value that you would expect to get over a large number of trials.
Example # 3:
A game is defined by the rules that two dice are rolled and the player wins varying
Amounts depending on the sum of the two dice rolled based on the following table:
Sum / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12Win / $10 / $9 / $8 / $7 / $6 / $5 / $6 / $7 / $8 / $9 / $10
If it costs $7.50 to play this game:
a) What can a player expect to win by playing this game?
Solution #3a:
Let the random variable X be the winnings associated with each roll.
Winnings($) / Sum / p(x) /
5 / 7 / /
6 / 6 or 8 / /
7 / 5 or 9 / /
8 / 4 or 10 / /
9 / 3 or 11 / /
10 / 2 or 12 / /
approx. $6.94
b) What would be the fair price to pay to play this game? (remember what fair means)
Solution #3b:
The fair price to play the game is $6.94 because at a price of $7.50 per game you lose $0.56 each time you play.
Example # 4:
A committee of four people is to be chosen randomly from four males and six females. What is the expected
number of females in the committee?
Solution #4:
Let the random variable X be the number of females on the committee.
X = { 0 , 1 , 2 , 3 , 4 }
Total # of committees
Total # of committees with;
0 girls 3 girls
1 girl 4 girls
2 girls
Probability Distribution# of females
(x) / p(x) / x * p(x)
0 / / 0
1 / /
2 / /
3 / /
4 / /
Therefore the expected number of females on the committee is 2.4
Pascal’s Triangle
Pascal’s Triangle is built with an interactive process. Each term in the triangle is equal to the sum of the two terms directly above it. The first and the last term in each row are both equal to 1.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
If tn , r represents the term in row (n) and in position (r), then tn , r = tn-1 , r-1 + tn-1 , r .
ie: t5 , 2 = t5-1 , 2-1 + t5-1 , 2 = t4 , 1 + t4 , 2 (10 = 4 + 6)
Example # 1:
The first six terms in row 25 of Pascal’s triangle are 1 , 25 , 300 , 2300 , 12 650 , and 53 130. Determine the first six terms in row 26?
Solution #1:
1 25 300 2300 12650 53130
1 26 325 2600 14950 65780
Example # 2:
Use pascal’s method to write a formula for each of the following:
a) t12 , 3 b) t40 , 32 c) tn+1 , r+1
Solution #2
a) b) c)
Example # 3:
Which row in Pascal’s triangle has the sum of its terms equal to 32 768?
Solution #3:
Row sum formula for Pascal’s Triangle is: Therefore, n =15
Example # 4:
Coins can be arranged in the shape of an equilateral triangle. Determine the number of coins in the triangle when there are four rows, five rows and six rows. Locate these numbers in Pascal’s Triangle. Relate Pascal’s triangle to the number of coins in a triangle with n rows. How many coins are there in a triangle with 12 rows?
Solution # 4:
Draw your piles of coins.
Total 1 3 6 10 15
Coins (2 x 2 x 2) (3 x 3 x 3) (4 x 4 x 4) (5 x 5 x 5)
Observe the pattern in Pascal’s Triangle.
Therefore, 12 rows has 78 coins. “Hockey Stick Pattern”
Example # 5: Counting Paths in an Array
Determine how many different paths will spell PASCAL, if you start at the top and proceed to the next row moving diagonally left or diagonally right.
P
A A
S S S
C C C C
A A A
L L
Solution #5
There is 20 ways of spelling the word “PASCAL”.
Example # 6:
Determine the number of ways the checker can move from its current position to the top of the board. The checker can only move diagonally upward and can not move through the square with the X.
X●
Solution #6:
8 / 8 / 84 / 4 / 4 / 4
4 / X / 4
1 / 3 / 3 / 1
1 / 2 / 1
1 / 1
●
Therefore there are 24 ways.
What would change if you were allowed to jump over the square with the X to the diagonally opposite side? 36
Binomial Theorem Day # 1
Consider the expansion of the following binomials.