Transcript Tree Diagrams and Sampling With/Without Replacement Owen Xia, Julian Lim, Nitin Shyamkumar and Joseph Myles.

Tree Diagrams and Sampling
With/Without Replacement
Owen Xia, Julian Lim, Nitin
Shyamkumar and Joseph Myles
What is a Tree Diagram?
• Tree diagrams map
out all the possible
possibilities as a
tree.
• This tree diagram
shows the
possibilities of
three coin tosses.
• For example, the
possibilities include
three heads, two
heads and one tail,
one head and two
tails and three tails.
Heads
Heads
Heads
Tails
Tails
Heads
Tails
Heads
Heads
Tails
Tails
Tails
Heads
Tails
50%
Heads
50%
Heads
50%
Heads
50%
Tails
12.5%
50%
Tails
12.5%
50%
Heads
12.5%
50%
Tails
12.5%
50%
Heads
50%
Heads
50%
Tails
50%
Tails
50%
Tails
12.5%
50%
12.5%
Heads
50%
Tails
•
•
12.5%
12.5%
In order to represent the probability of each outcome, we multiply the probability of each
independent or dependent event that’s happening.
For example, the possibility of getting three heads is equal to the product of the three outcome
probabilities. Since the chance of getting heads is an independent event, it always has a chance of
50%. Therefore, the chance of getting three heads is 12.5%.
What does it mean?
Chance of Chance of 2 Chance of 1 Chance
3 Heads Heads, 1 Tail Head, 2 Tails of 3 Tails
12.50%
37.50%
37.50% 12.50%
•The probabilities of all the outcomes are expressed in the above table.
The probabilities of the same outcome are added together and are thus
expressed.
•The sum of all probabilities will always equal 1.
•In this situation, the multiple coin tosses are independent events. Later,
we will investigate dependent events.
Problem
• If you roll two die,
what is the probability
of receiving a sum of at
least 8? Make a tree
diagram and find the
probability.
• Answer: There is a
15/36 or 5/12
possibility of receiving
a sum of at least 8.
Tree Diagram of Answer
(Red notates every outcome
with a sum greater than 8)
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
Every outcome has a 1/36
chance of occurring. There
are 15 outcomes with a sum
of at least 8.
Therefore, the chance of
getting a sum of at least 8 is
15/36
We will now explain
dependent events.
Dependent Events
•
A dependent event is an event whose outcome
can be influenced by a preceding event. The most
common type of dependent event is “without
replacement”
•
Example: Shim Dragon has to randomly pick two
students to compete in the All Valley Math
Tournament and Nitin really wants to go.
•
Shim Dragon’s chances of picking Nitin the first
time are 1/30. If she does not pick Nitin at first, he
has a 1/29 chance to be picked because there are
only 29 other kids to pick from, as opposed to 30.
•
You can find the probability of Nitin getting picked
by adding both probabilities, resulting in a 59/870
chance of him being able to go to the All Valley
Math Tournament.
Dependent Events with Tree Diagrams
• Like independent events, dependent events can also be
expressed in tree diagrams.
• Example: If Joe reaches into a bag containing 12 blue
marbles and 17 red marbles and pulls out two marbles,
what are the chances of him getting one blue and one red?
16/28
17/29
Red
Red
12/28
Blue
17/28
12/29
Blue
Red
272/812
204/812
204/812
11/28
Blue
132/812
There is a
102/203 chance
of getting one
blue and one red
With/Without Replacement
• The problem we just did is
implied to be “without
replacement.”
• Therefore, the term “with
replacement” ensures
independent events and the
term “without
replacement” means
dependent events.
• In the IB Exam, it will either
be explicitly stipulated or
implied whether or not the
events are independent or
dependent.
Final Problem: There is a box of
6 sprinkled donuts, 6 jelly
donuts and 12 glazed donuts.
If Albert randomly picks three
donuts without replacement
(because he loves donuts), what
are his chances of picking
exactly two jelly donuts?
If Albert randomly picks three
donuts with replacement
(because likes to touch other
people’s food), what are his
chances of picking a glazed
donut first and then picking two
sprinkled donuts?
HL Book: 18F 3-9 Odds