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Created by: Aman Momin
GSMST – Accelerated Geometry
UNIT 4: PROBABILITY
INTRODUCTION TO SETS
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Set – Contains an object or objects that are clearly identified
The objects in a set are called elements or members.
o Example 1: 3 ϵ { 1,2,3,4 }
o Example 2: 8 { 1,2,3,4 }
A complement of a set is everything that does NOT belong to the set.
o If there exists a set named A then the complement is denoted by A’ or A
o A set can be empty and denoted by Ø
 Example: {living unicorns} = Ø
SUBSETS
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If every member in set A is also a member of set B, then A is a subset of set B
⊆ means “is a subset of”
⊄ means “is not a subset of”
INTERSECTION OF SETS
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To find the intersection of, two sets, you need to identify the members that the two sets share in common
∩ is the symbol for intersection, Means “and”
Example
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{2,4,6,8} ∩ {4,6,8,10} = {4,6,8}
A Roster shows a member of each set
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Created by: Aman Momin
GSMST – Accelerated Geometry
UNION OF SETS
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Means “or”
Example:
PROBABILITY
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Probability is the measure of how likely an event is to occur.
Each possible result of a probability experiment or situation is an outcome.
The sample space is the set of all the possible outcomes.
An event is an outcome or set of outcomes.
Equally likely outcomes have the same chance of occurring.
o Example: When you toss a coin, heads and tails are equally likely outcomes.
Each repetition of an experiment is a trial
THEORETICAL PROBABILITY
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Theoretical Probability of an event is the ratio of the number of favorable outcomes to the total number
of outcomes.
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𝑃(𝑒𝑣𝑒𝑛𝑡) =
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Example:
o A red number cube and a blue number cube are rolled. If all numbers are equally likely, what is
the probability that the sum is 10?
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑤𝑖𝑡ℎ 𝑠𝑢𝑚 𝑜𝑓 10
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3
36
o
𝑃(𝑠𝑢𝑚 𝑖𝑠 10) =
o
𝑃(𝑠𝑢𝑚 𝑖𝑠 10) =
o
Answer: 3 Outcomes
EXPERIMENTAL PROBABILITY
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The experimental probability of an event is the ratio of the number of times that the event occurs, the
frequency, to the number of trials.
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𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠
2
Created by: Aman Momin
GSMST – Accelerated Geometry
INDEPENDENT AND DEPENDENT EVENTS
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Events are independent events if the occurrence of one event does not affect the probability of the other.
If a coin is tossed twice, its landing heads up on the first toss and landing heads up on the second toss are
independent events. The outcome of one toss does not affect the probability of heads on the other toss.
To find the probability of tossing heads twice, multiply the individual probabilities.
If A and B are independent events, then P(A and B) = P(A) x P(B)
Events are dependent events if the occurrence of one event affects the probability of the other.
If A and B are dependent events, the P(A and B) = P(A) x P(B | A), where P(B | A) is the probability of B,
given that A has occurred.
CONDITIONAL PROBABILITY
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Conditional Probability is the probability of an event (A), given that another (B) has already occurred.
Formula for Conditional Probability
o
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𝑃(𝐴|𝐵) =
P( A ∩ B)
𝑃(𝐵)
Probability of Independent Events A and B
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𝑃(𝐴) =
P( A ∩ B)
𝑃(𝐵)
o
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For independent events, the probability of two events occurring together is simply the product of
each event’s individual probability
P (A | B) asks that we find the probability of A given that we know B has or already occurred. Using a
formula find the probability of A given B can be found using 𝑃(𝐴) =
P( A and B)
𝑃(𝐵)
COMPOUND EVENTS
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A compound event is an event made up of two or more simple events.
Mutually exclusive events (or exclusive events) are events that cannot both occur in the same trial of an
experiment.
o Algebra: 𝑃(𝐴 ∪ B) = 𝑃(𝐴) + 𝑃(𝐵)
o Example:
 When a number cube is rolled, P(less than 3) = P(1 or 2) =
1
1
1
6
6
3
P(1) + P(2) = + =
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Inclusive events are events that have one or more outcomes in common.
o Algebra: 𝑃(𝐴 ∪ B) = 𝑃(𝐴) + 𝑃(𝐵) − P( A ∩ B)
o Example:
 When you roll a number cube, the outcomes “rolling an even number” and “rolling a
prime number” are not mutually exclusive. The number 2 is both prime and even so the
events are inclusive.
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3
3
1
5
6
6
6
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P(even) + P(prime) – P(even and prime) = + − =
3