IB Math SL Year 2
Name: ___________________________
Date: __________________
2-8 Notes: Independent Probability, Conditional Probability, Tree Diagrams
In today’s Lesson we will address the following learning goals:
1) What does it mean for events to be independent and how do we determine if two events are independent?
2) What is conditional probability? How do we determine conditional probabilities?
3) How do we create a tree diagram? When should we use a tree diagram to determine probabilities?
Use your formula booklet!
What page is this on in your formula booklet?? _____________
Station 1: Sample Space
Fact Check:
Sample space is a list of all possible outcomes
Universal Set set of all possible outcomes (Notation is U)
*Always use curly brackets { …} when listing elements of a set
Example) A die
Guided Example:
1. In an experiment a coin is tossed and a die is rolled.
a. Draw the sample space diagram for this experiment.
b. Hence find the probability that in a single experiment you obtain a head and a number less than 3 on the
die.
IB Math SL Year 2
Check out of Station 1
Complete this problem set. Check in with your teacher to receive your next station
1. A box contains three cards bearing the numbers 1,2, 3. A second box contains four cards with numbers 2,3,4,5.
A card is chosen at random from each box.
a) Draw the sample space diagram for the random experiment.
Find the probability that:
b) The cards have the same number
c) The larger of the two numbers drawn is 3
d) The sum of the two numbers on the cards is a less than 7
e) The product of the numbers on the cards is at least 8
f)
At least one even number is chosen
IB Math SL Year 2
Station 2: Independent Events
Fact Check
Two events are Independent if the occurrence of one does not affect the occurrence of the other.
In terms of mathematics this means we can _____________________________ probabilities!
Other ways to say independent: not associate, not related
On your Formula Page:
Guided Examples:
1. Let P(A) = 0.6, P(B) = 0.5 and 𝑃(𝐴 ∩ 𝐵) = 0.2. Are A and B independent? Justify your answer with mathematics.
2. An urn contains 3 red, 4 yellow and 5 black marbles. Three marbles are drawn at random without replacement.
Find the probability a red is drawn first, a yellow next, and a black is drawn last.
IB Math SL Year 2
Check out of Station 2
Complete this problem set. Check in with your teacher to receive your next station
4
1. A large school conducts a survey of the food provided by the school cafeteria. It was found that 5 of
the students like pasta. Three students are chosen at random. What is the probability that all three
students like pasta?
2. Consider events E and F where P(E) = .4, P(F) = .5, and P(E ∩ F) = .2 Are E and F independent events? Show
math to justify your answer.
3. A school has two photocopiers. On any one day, machine A has an 8% chance of malfunctioning and machine B
has 12% chance of malfunctioning.
Determine the probability that on any one day both machines will:
a. Malfunction
b. work effectively
IB Math SL Year 2
Station 3: Conditional Probability
Conditional ProbabilityAs a picture**NEW** Notation
To find the probability of the event A given the event B, we restrict our
attention to the outcomes in B. We then find the fraction of those
outcomes A that also occurred.
From the formula sheet
This becomes….
-
Starting points for problems, when determining conditional probabilities:
If given the probabilities, just use the formula
If you have a Venn Diagram, circle the set you are given
If given a table, circle the column/row of what you are
and that is your denominator
given and use that as your denominator
If given data fill into either a table or a Venn Diagram
Guided Examples
2
2
1. Given that 𝑃(𝐶) = 3 𝑎𝑛𝑑 𝑃(𝐶 ∩ 𝐷) = 5 , 𝑓𝑖𝑛𝑑 𝑃(𝐷|𝐶).
2. Of the 53 staff at a school, 36 drink tea, 18 drink coffee, and 10 drink neither tea nor coffee.
Space for Venn Diagram
a. How many staff drink both tea and coffee?
One member is chosen at random. Find the probability that:
b. he drinks tea but not coffee,
c. he drinks coffee given that he drinks tea,
d. he does not drink coffee given that he drinks tea.
IB Math SL Year 2
Check out of Station 3
Complete this problem set. Check in with your teacher to receive your next station
1. There are 27 Students in a class. 15 take art, 20 take theater and four take neither subject. How many
students take both subjects?
One person is chosen at random. Find the probability that he or she:
a) takes theater but not art
b) Takes at least one of the two subjects
c) Take theater given that he or she takes art
2. For events A and B it is known that P(A’∩ 𝐵′) = 0.35; 𝑃(𝐴) = 0.25; 𝑃(𝐵) = 0.6. 𝐹𝑖𝑛𝑑:
a) P(A∩B)
b) P(A|B)
c) P(B’|A’)
3. 48% of all teenagers own a skate board and 39% of all teenagers own a skateboard and roller blades. What
is the probability that a teenager owns roller blades given that the teenager owns a skateboard?
IB Math SL Year 2
Station 4: Tree diagrams
1.
When working with tree diagrams always remember to:
Always fill in all the values on your branches
2.
Multiply across the branches!
3.
Use these products to get your answer! NEVER use the values on the branches!
Let’s Try it!
1. A teacher has a box containing six type A calculators and four type B calculators. The probability that a
type A calculator is faulty is 0.1 and the probability that a type B calculator is faulty is 0.12.
(a)
Complete the tree diagram given below, showing all the probabilities.
0.1
Probability outcomes
FAULTY
type A
0.6
NOT FAULTY
FAULTY
0.4
type B
NOT FAULTY
(b)
A calculator is selected at random from the box. Find the probability that the calculator is
(i)
not faulty.
(ii)
Type B, given it was not faulty.
IB Math SL Year 2
2. At the basketball game, Amanda got into a two-shot foul situation. She figured her chance of making the first
shot was 0.7. If she made the first shot, her chance of making the second shot was 0.6. However, if she missed
the first shot, her probability of making the second shot was only 0.4.
a)
Complete the Tree Diagram above.
b)
What is the probability Amanda misses the second shot.
c)
Given Amanda missed the second shot, find the probability that she made the first shot.
IB Math SL Year 2
Check out of Station 4
Complete this problem set. Check it with your teacher to receive your hw!
3
. If the weather is good, the probability he
4
17
1
will go walking (W) is
. If the weather forecast is not good (NG) the probability he will go walking is .
20
5
(a)
Complete the probability tree diagram to illustrate this information.
Today Philip intends to go walking. The probability of good weather (G) is
W
17
20
G
3
4
NW
W
NG
NW
(b)
What is the probability that Philip will go walking?
(c)
What is the probability that Philip will go walking given that it is good weather?
IB Math SL Year 2
Name:
HW
Date
Complete each of the following problems. Show work and thinking for each. Use the formula booklet to help you
through the problems. Don’t Forget to Check the key tonight!
1. I toss a coin and roll a six-sided dice. Find the probability that I get a head on the coin and don’t get a 6 on the
dice?
2. Given that P(B) = 0.8 and P(A∩B) = 0.6. Find P(A|B).
3. My wardrobe contains 7 shirts with one blue, one brown, two red, one white and two black. I reach into the
wardrobe and choose a shirt without looking to wear for work. Then I grab another shirt from the wardrobe for
my cousins birthday dinner after work What is the probability that I will choose a red shirt both times?
4. In the real estate ads, 64% of homes have garages, 21% have swimming pools, and 17% have both features.
a) What’s the probability a home has a garage or pool?
Are taking these two events independent events? Justify with mathematics.
IB Math SL Year 2
1. In a survey of 200 people, 90 of whom were female, it was found that 60 people were unemployed, including 20
males.
a. Using this information, complete the table below.
Males
Females
Totals
Unemployed
Employed
Totals
200
b. If a person is selected at random from this group of 200, find the probability that this person is
i. an unemployed female;
ii. a male, given that the person is employed.
3. Jim drives to work each day through two sets of traffic lights. The probability of the first set of traffic lights
being red is 0.65. If the first set is red then the probability that the next set of traffic lights is red is 0.46. If the
first set is not red, the probability that the next set is red is 0.72.
red
red
0.65
not
red
red
not
red
not
red
a)
b)
Complete the tree diagram above.
Calculate the probability that the second set of traffic lights is red
c)
Calculate the probability that the first light is not red, given the second set of traffic lights is red
IB Math SL Year 2
1. Researchers analyzed 200 adults to determine if there was a link
between their highest level of education and whether or not they
smoked. Use the table at the right to answer the following
questions, leave all questions in fraction form and use proper
notation.
a) Given the person smoked, what's the probability they graduated
high school?
b) Given the person smoked, what's the probability they went to a 2 year college?
1. Let P(A) = 0.6, P(B) = 0.5 and 𝑃(𝐴 ∩ 𝐵) = 0.2
a) Draw a Venn diagram to represent the given information
Find each of the following probabilities
b) 𝑃(𝐴 ∪ 𝐵)
c) 𝑃((𝐴 ∪ 𝐵)′ )
d)𝑃(𝐴|𝐵)
e)𝑃(𝐵|𝐴)