8.1-8.
Probability
Julianna,Karina Joy, Madison, Sitian, Adheena
What is
Probability?
How likely something is to happen
What are
statistics?
The analysis of events controlled by
probability
weather
medicine
Probability of Compound events
Definition : A compound event is an event that combines two or more simple events.
Key Terms :
•Single Event: An event with a single outcome (e.g., rolling a 4 on a die).
•Compound Event: Combines simple events using “AND” or “OR.”
•Mutually Exclusive Events: Events that cannot occur at the same time.
•Addition Rule: For mutually exclusive events, P(A \text{ or } B) = P(A) + P(B)
Examples:
Rolling a die: P(rolling 1 or 2) =P(1) +P(2)=⅙+⅙=2/6
Tossing a coin twice: P(at least one head)=P(HH)+P(HT)+P(TH)=0.25+0.25+0.25=0.75
P (E) = number of favorable outcomes/total number of possible outcomes
Single stage experiments - experiments that end after one step. Ex: rolling a die, flipping a coin
Ex: There are 4 cards, a circle, heart, square and oval. If one card is randomly
selected, how likely is it to obtain a card with a heart?
If the card with the heart is removed from the deck, what is the probability to
obtain a card with a square?
Ex: The probability that the roads will be closed on Saturday is 0.6. The probability
that it snows on Saturday is 0.3. What is the probability that the roads are closed
AND it snows this Saturday?
Disjoint things - they cannot occur at the same time
A∩B=0
Law of Large Numbers : This law states that the more times a simulation is carried out,
the closer the experimental probability is to theoretical probability.
Odds : pair of numbers often used in gambling and stats
Odds for : the likelihood the event will happen in comparison to not happening
Odds against : the likelihood the event will NOT happen in comparison it will happen
Ex:
Favorable Unfavorable Odds for = 1 to 3
Odds against = 3 to 1
Probability of an Event Formula
When creating the probability event
formula, you would have to place the
possibility of outcomes on the top part
of the fraction. ( the outcome you are
looking for) and the bottom part has the
total amount of possible outcomes.
(total amount of given value)
Example
Example: A cookie jar contains 3 chocolate chip cookie,
2 raisin cookies and 1 sugar cookie. If I closed my eyes
and chose a cookie. What is the probability that I will
get a chocolate chip cookie?
Experimental Probability
The Experimental Probability has to do with
how many times something happened in
the past. Also the total amount of trials that
you made during that experiment. The
result of an experiment
Do it yourself: A bottle of water was flipped
20 times. It landed 15 times. In the other
hand it fell 5 times. Using the experimental
probability formula how would you write
down the probability of this given event?
Theoretical Probability
Theoretical Probability has to do
with the expected outcome of an
experiment, It goes off of the possible
number of outcomes in the given
situation, as well as the total number of
possible outcomes.
Example (Together): If we have a
spinner with 2 red sections, 1 yellow
section and 3 blue sections. What is
the possibility for it to land on a
red?
Probability Glossary
Experiment: An experiment is the event
that we are observing and analyzing the
given results provided
Outcome: An outcome is the results
gained after the experiment happens
Event: Event is a set of result/ outcomes
from the experiment
Sample Space: A sample Space is the set
with all the possible given outcomes from
the experiment.
Examples
Sample Space: If you rolled a 7 sided die once and obtained a 3. Determine
the sample space and the probability of the given outcomes?
SS 1,2,3,4,5,6,7
P( z)= n(3)/ n(1,2,3,4,5,6,7,)= 1/7
1 2 3 4 5 6 7
Probability of Event
★ The probability of any event is always less than or equal to 1.
0 ≤ P(E) ≤ 1
A bag containing eight scraps of paper, each marked with a number.
P(1) =
P(2) =
P(odd) =
P(even) =
P(9) =
P(less than 9) =
Example:
A bag containing eight scraps of paper, each marked with a number.
P(1) =
P(2) =
P(odd) =
P(even) =
P(9) =
P(less than 9) =
Example:
A bag containing eight scraps of paper, each marked with a number.
P(1) =
P(2) =
P(odd) =
P(even) =
P(9) =
P(less than 9) =
Example:
A bag containing eight scraps of paper, each marked with a number.
P(1) =
P(2) =
P(odd) =
P(even) =
P(9) =
P(less than 9) =
Example:
A bag containing eight scraps of paper, each marked with a number.
P(1) =
P(2) =
P(odd) =
P(even) =
P(9) =
P(less than 9) =
Example:
A bag containing eight scraps of paper, each marked with a number.
P(1) =
P(2) =
P(odd) =
P(even) =
P(9) =
P(less than 9) =
Example:
A bag containing eight scraps of paper, each marked with a number.
P(1) =
P(2) =
P(odd) =
P(even) =
P(9) =
P(less than 9) =
Example:
A bag containing eight scraps of paper, each marked with a number.
P(1) =
P(2) =
P(odd) =
P(even) =
P(9) =
P(less than 9) =
Impossible Event Vs. Certain Event
P(E) =0 P(E) = 1
-Not likely to happen -Will happen
Chapter 8.2
Dependent Events
★ Two events are dependent when the occurrence of one influences the
probability of the other.
Example:
In a deck of 52 playing cards, you draw one card without replacing it and
then draw another card. What is the probability that the first card is a
King and the second card is a Queen?
Example :
★ In a deck of 52 playing cards, you draw one card without replacing it and then draw
another card. What is the probability that the first card is a King and the second
card is a Queen?
Step 1: Find the probability of drawing a King
on the first draw. There are 4 Kings in a deck
of 52 cards, so the probability is:
Example :
★ In a deck of 52 playing cards, you draw one card without replacing it and then draw
another card. What is the probability that the first card is a King and the second
card is a Queen?
Step 1: Find the probability of drawing a King
on the first draw. There are 4 Kings in a deck
of 52 cards, so the probability is:
Example :
★ In a deck of 52 playing cards, you draw one card without replacing it and then draw
another card. What is the probability that the first card is a King and the second
card is a Queen?
Step 1: Find the probability of drawing a King
on the first draw. There are 4 Kings in a deck
of 52 cards, so the probability is:
Example :
★ In a deck of 52 playing cards, you draw one card without replacing it and then draw
another card. What is the probability that the first card is a King and the second
card is a Queen?
Step 2: Find the probability of drawing a Queen on
the
- second draw, given that the first card was a
King.
After removing a King, only 51 cards remain in the
deck, and there are still 4 Queens
Example :
★ In a deck of 52 playing cards, you draw one card without replacing it and then draw
another card. What is the probability that the first card is a King and the second
card is a Queen?
Step 2: Find the probability of drawing a Queen on
the
- second draw, given that the first card was a
King.
After removing a King, only 51 cards remain in the
deck, and there are still 4 Queens
Given that Kings happened
Example :
★ In a deck of 52 playing cards, you draw one card without replacing it and then draw
another card. What is the probability that the first card is a King and the second
card is a Queen?
Step 3: Multiply the probabilities of the dependent
events. The probability of both events happening
(drawing a King first and a Queen second) is:
Example :
★ In a deck of 52 playing cards, you draw one card without replacing it and then draw
another card. What is the probability that the first card is a King and the second
card is a Queen?
Step 3: Multiply the probabilities of the dependent
events. The probability of both events happening
(drawing a King first and a Queen second) is:
Example :
★ In a deck of 52 playing cards, you draw one card without replacing it and then draw
another card. What is the probability that the first card is a King and the second
card is a Queen?
-
Addition Property
★ For events A and B, the probability of A or B or both occurring is:
P (A U B) = P(A) + P(B) - P(A N B)
★ If A and B are independent events, then:
P(A U B) - P(A) + P(B) - P(A) * P(B)
Example: A die is rolled. What is the probability of rolling a number greater than 4 or
an even number?
Example
A die is rolled. What is the probability of rolling a number greater than 4 or an even
number?
Step 1:
Define the
sample
space:
The possible
outcomes
when rolling
a die are:
{1,2,3,4,5,6}
Example
A die is rolled. What is the probability of rolling a number greater than 4 or an even
number?
Step 1: Step 2: Identify the
Define the events:
sample
space: Event A : Rolling a
The possible number greater than 4 =
outcomes {5, 6}
when rolling
a die are:
{1,2,3,4,5,6} Event B : Rolling an even
number = {2, 4, 6}
Example
A die is rolled. What is the probability of rolling a number greater than 4 or an even
number?
Step 1: Step 2: Identify the
Define the events:
sample
space: Event A : Rolling a
The possible number greater than 4 =
outcomes {5, 6}
when rolling
a die are:
{1,2,3,4,5,6} Event B : Rolling an even
number = {2, 4, 6}
Example
A die is rolled. What is the probability of rolling a number greater than 4 or an even
number?
Step 1: Step 2: Identify the Step 3: Find the
Define the events: overlap
sample (intersection):
space: Event A : Rolling a
The numbers
The possible number greater than 4 =
greater than 4
outcomes {5, 6}
that are also
when rolling even = {6}
a die are:
{1,2,3,4,5,6} Event B : Rolling an even
number = {2, 4, 6}
Example
A die is rolled. What is the probability of rolling a number greater than 4 or an even
number?
Step 1: Step 2: Identify the
Define the events:
sample
space: Event A : Rolling a
The possible number greater than 4 =
outcomes {5, 6}
when rolling
a die are:
{1,2,3,4,5,6} Event B : Rolling an even
number = {2, 4, 6}
Example
A die is rolled. What is the probability of rolling a number greater than 4 or an even
number?
Step 4: Apply the Addition
Step 1: Step 2: Identify the
Property:
Define the events:
The probability of rolling a
sample
space: Event A : Rolling a number greater than 4 or an
The possible number greater than 4 = even number is given by:
outcomes {5, 6}
when rolling
a die are: Substituting the probabilities:
{1,2,3,4,5,6} Event B : Rolling an even
number = {2, 4, 6}
Find a common denominator
and simplify:
Example
A die is rolled. What is the probability of rolling a number greater than 4 or an even
number?
Step 1: Step 2: Identify the
Define the events:
sample
space: Event A : Rolling a
The possible number greater than 4 =
outcomes {5, 6}
when rolling
a die are:
{1,2,3,4,5,6} Event B : Rolling an even
number = {2, 4, 6}
Example
A die is rolled. What is the probability of rolling a number
greater than 4 or an even number?
Solution: The probability of rolling a number greater than
4 or an even number is
8.2 Test for Independence
𝑷 (𝑨 ∩ 𝑩) = P (A) x P (B)
This means these two events are independent - one outcome doesn’t influence the other
Ex: if you flip a coin (heads or tails) the first outcome will not affect the second outcome
Complementary events - The probability of an event can be most easily found by
calculating the probability of the complement
Ex: If a six sided die is tossed 4 times, what is the probability of obtaining at least one 6?
Thank you!
