Download IB Probability 9C-E

10/27/2015
Coin Experiment
When we toss a coin in the air, we expect it to finish on a head or tail with equal
likelihood.
What to do:
• Toss one coin 40 times. Record the number of heads in each trial, in a table:
Toss two coins 60 times. Record the number of heads in each trial, in a table.
Toss three coins 80 times. Record the number of heads in each trial, in a table.
• Share your results for 1, 2, and 3 with several other
students. Comment on any similarities and differences.
• Pool your results and find new relative frequencies
for tossing one coin, two coins, and three coins.
•Be ready to share your results with the class.
Questions on the homework?
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PROBABILITY CONTINUED
Learning Target 3.2
Theoretical Probability
Learning Target 3.3
Compound events
C. THEORETICAL PROBABILITY
The likelihood of obtaining the outcome 2 would be:
The likelihood of obtaining the outcome 2 or 5 would be:
This is a mathematical or theoretical probability and it is based on what we
theoretically expect to occur. It is the chance of that event occurring in any
trial of the experiment. Theoretically, a coin toss showing heads should occur 1
every 2 times thrown.
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SO… HOW DO QUANTIFY IT
“A” is the event. 𝑈 is the sample space that event can occur in. So 𝑃(𝐴) is
the probability that that event can occur in the sample space 𝑈.
EXAMPLE
•
You have a bag of Jelly Beans and you randomly selected from the bag. It
contains:
• 15
Buttered Popcorn
• 7
Coffee
• 9
Boogers
•
1) Determine the probability of getting the booger jellybean?
•
2) What is the probability of not getting the coffee one?
•
3) What is the probability of getting a booger one or a coffee one?
•
4) what is the probability of getting a grape one?
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YOU TRY ON YOUR OWN. STOP TALKING. GRACE.
•
An ordinary 8 sided die is rolled once. Determine the chance of:
A) Getting a 6
B)
𝑃 𝑛𝑜𝑡 6
C)
Getting a 1 or 2
D)
Not getting a 1 or 2.
COMPLEMENTARY EVENTS
• Two events are complementary if exactly one of the events must
occur. If A is an event, then A’ is the complementary event of A, or
‘not A’.
• 𝑷(𝐴) + 𝑷(𝐴′) = 1
• This reads, the probability of A occurring plus the probability of A
not occurring equals 1.
• We can solve for each of the probabilities.
• 𝑃 𝐴 = 1 − 𝑃(𝐴′ )
• 𝑃 𝐴′ = 1 − 𝑃 𝐴
• Example: Emma is going to the park. She has seen her friends at
the park hanging out and rollerblading 23% of the time. What is
the probability that Emma will not see her friends?
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ANOTHER EXAMPLE!
•
Use a 2D grid to illustrate the sample space for tossing a coin and rolling a 6
sided die simultaneously. From this grid, determine the probability of:
a) Tossing a head
b) Getting a Tail and a 5
c)
Not Getting a tail or a 6.
EXAMPLE 3
•
Draw a table of outcomes to display the possible results when two dice are
rolled and the scores are added together.
•
Determine the probability that the sum of the dice is 7.
•
Find the probability that the dice will not add to be 8.
6
7
8
9
10
11
12
5
6
7
8
9
10
11
4
5
6
7
8
9
10
3
4
5
6
7
8
9
2
3
4
5
6
7
8
1
2
3
4
5
6
7
1
2
3
4
5
6
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D. COMPOUND EVENTS
•
Consider the following problem:
•
Box X contains 2 blue and 2 green balls. Box Y contains
1 white and 3 red balls. A ball is randomly selected
from each of the boxes. Determine the probability of
getting “a blue ball from X and a red ball from Y”
•
What can we say about these two events happening?
What do we do to interpret the 2-D grid?
W
R
R
R
B
B
G
G
D. INDEPENDENT EVENTS
•
•
We can see that the area highlighted is the possible outcomes and the total area
is the universal set.
6
16
•
We can do this without drawing a 2D grid
because they are independent compound events.
•
two events for which the occurrence of each one
does not affect the occurrence of the other,
we call these independent events.
•
𝑃 𝐵 𝑎𝑛𝑑 𝑅 = 𝑃 𝐵 ∙ 𝑃 𝑅 .
W
•
1
2
3
4
=
3
8
R
R
R
B
B
G
G
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D. INDEPENDENT EVENTS EXAMPLE
•
Franco went to the store to buy some vegetables for his pet rabbit. Before he
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went he found out that carrots were dirty and may cause his rabbit to get
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2
sick. He also discovered that lettuce heads are dirty as well. Assuming that
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he gets equal amounts of each vegetable;
A) What are the chances that Franco will pull a dirty carrot and dirty lettuce?
B)
What are the chances of Franco not pulling a dirty vegetable?
D. DEPENDENT EVENTS
•
A dependent event happens when one even occurs that affects the following
event.
•
If event “A” effects event “B”, we would write:
•
𝑃 𝐴 𝑡ℎ𝑒𝑛 𝐵 = 𝑃 𝐴 ∙ 𝑃 𝐵, 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑎𝑡 𝐴 ℎ𝑎𝑠 𝑜𝑐𝑐𝑢𝑟𝑟𝑒𝑑
•
Example:
•
Suppose an urn contains 5 red and 3 blue tickets. One ticket is randomly
chosen, its colour is noted, and it is then put aside. A second ticket is then
randomly selected.
•
If the first ticket was red, what would the probability the second is red?
•
If the first ticket was blue, what would the probability be second is red?
What is the probability that the first one was red and the second one is blue?
5
8
3
15
=
7
56
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E. USING A TREE DIAGRAM
• Kareem is not having much luck with his motor
vehicles. His car will only start 80% of the time
and his motorcycle will only start 60% of the
time.
• Are these dependent or independent events?
• Construct a tree diagram to illustrate his
situation.
• Use the diagram to determine the chance that
• Both will start.
• Kareem can only use his car.
First
start
MC
start
Car
M
M’
Car’
M
M’
ADDING EVENTS
•
If there is more than one outcome in an event then we need to add the
probabilities of these outcomes.
Example
•
Two boxes each contain 6 petunia plants that are not yet flowering. Box A
contains 2 plants that will have purple flowers and 4 plants that will have white
flowers. Box B contains 5 plants that will have purple flowers and 1 plant that will
have white flowers. A box is selected by tossing a coin, and one plant is removed
at random from it. Determine the probability that it will have purple flowers using
a tree diagram.
•
We are trying to find 𝑃(𝑝𝑢𝑟𝑝𝑙𝑒 𝑓𝑙𝑜𝑤𝑒𝑟𝑠),
therefore, this equals 𝑃(𝐴 𝑎𝑛𝑑 𝑃) + 𝑃(𝐵 𝑎𝑛𝑑 𝑃)
•
You complete the exercise.
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HW: 9C-E
•
C.1 #2-4
•
C.2 #2
•
C.3 #1,3
•
D.1 #1,2,6
•
D.2 #1,2,5
•
E #1,2,5-7
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