Download FD12 5.1 Lesson KEY

Foundations of Mathematics 12
KIM
Ch5. Probability
Probability is a theory that deals with the mathematics of chance or prediction. In this unit, we will
learn to develop critical thinking skills related to uncertainty.
5.1 Exploring Probability
Goal: Today, we will learn terminology involving probability and use probability to make predictions.
Review:
a) A fair die is rolled. What is the probability of rolling a “one”?
b) A circular spinner is divided into four equal sectors, labelled clubs, diamonds, hearts, and
spades. When it is spun, what is the probability that it lands on “diamonds”?
Terminology
 Trial – any operation whose outcome cannot be predicted with certainty
e.g. toss a coin; roll a die
 Experiment – consists of one or more trials
e.g. roll a die; toss 2 coins
 Outcome – the result of carrying out an experiment
e.g. 6; TT
 Sample space – the set of all possible outcomes
e.g. {1,2,3,4,5,6}; {TT, TH, HH}
Foundations of Mathematics 12
KIM
 Event – a subset of the sample space (one or more of the possible outcomes)
e.g. {1,2,5}; {TH}
probability of event A =
*Impossible event – if even X does not include any of the outcomes in the
sample space
e.g. roll a 9 on a die
P(x) = 0
*certain event – if even Y includes ALL the outcomes in the sample space
e.g. roll a number less than 10
P(Y) =1
 Fair experiment – all the possible outcomes are equally likely
P(A) = # of outcomes favourable to A
Total # of possible outcomes
e.g. p(roll a prime number on a die)
 Complementary Event – the set of outcomes in the sample space that are not
included in the outcomes of event A
e.g. draw a heart & don’t don’t draw a heart
P(A) + P(A’) = 1
Foundations of Mathematics 12
KIM
Ex.1 A circular spinner is divided into four equal sectors, labelled clubs, diamonds,
hearts and spaces. When it is spun, what is the probability it lands on hearts?
Determine:
a) the sample space: {clubs, diamonds, hearts, spaces}
b) the favourable outcomes in the sample space: Hearts
c) if the outcome is equally likely: YES since the spinner is divided into equal sectors
d) the probability of the event: 1/4
Ex.2 What is the probability that in a family of 3 children there are exactly 2 girls?
Steps:
Draw a tree diagram to
present the possible outcomes
Ask if all outcomes are equally likely
Determine the number of branches
that contain the favourable outcome and divide
this by the total number of branches
=3/8
Foundations of Mathematics 12
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Ex. 3 Consider an experiment of spinning an equally spaced triangular spinner
numbered 1,2,3 and tossing two coins. State the probability of obtaining:
a) a three and two heads
1/12
b) a prime number and exactly one tail
4/12 =1/3
Ex. 4 A blue die and red die are rolled. The outcome “3 on the blue die and 4 on the
red die” can be represented by the ordered pair (3,4).
a) Show all the possible outcomes in the table provided.
Red
1
2
3
4
5
6
Blue
1
2
3
(3,4)
4
5
6
b) State the probability of the following events:
i) the same number appears on both dice. 1/6
ii) a different number appears on each dice. 5/6
iii) how can the answer for ii) be determined using the answer in i)? complement
Foundations of Mathematics 12
Practice Problems
1. What is the probability of rolling a fair die and getting a 3?
2. What is the probability of rolling a fair die and getting an even number?
3. What is the probability of rolling a fair die and getting a number from 1 to 6?
4. What is the probability of rolling a fair die and getting a 7?
5. What is the probability of drawing a face card from a shuffled deck of cards?
6. What is the probability of drawing a red ace from a shuffled deck of cards?
7. What is the probability of drawing a black card from a shuffled deck of cards?
8. What is the probability of drawing a non-face card from a shuffled deck of cards?
9. What is the probability of drawing a ten of spaces from a shuffled deck of cards?
10. Consider class example, Ex. 3. Use the tree diagram to answer the following:
a) Determine the probability of there being exactly 1 boy in a family of 3 children.
b) Determine the probability that the first child is a boy and the third child is a girl.
11. When the pointer on the left is spun twice, find the following probability:
a) the sum of the numbers is 6.
b) the same number comes up on each spinning.
Draw a table diagram to answer.
Answer: 1. 1/6, 2. ½, 3.1, 4. 0, 5. 3/13, 6. 1/26, 7.1/2, 8.10/13, 9. 1/52, 10 a)3/8, b)1/4, 11a) 5/64, b)1/8
KIM