Download Probability - Homework Market

Probability
Definition:
Probability: the chance an event will happen.
Probability =
# of ways a certain event can occur
# of possible events
 Probability must be a value between 0 and 1.
 The probability of the set of all possible outcomes of a trial must be 1
 The probability of an event occurring is 1 minus the probability that it does not
occur.
Complement of A
(Ac )
A
Rule for Complementary Events:
P (A) + P(Ac)= 1
Example:
If the probability that a person lives in an industrialized country of the world is 1/5,
find the probability that a person does not live in an industrialized country.
Answer:
P (not living in an industrialized country) = 1- P (living in an industrialized
country)= 1 – 1/5= 4/5
Simple Probability
Example 1:
What is the probability of rolling a 6 on a six sided die?
Know:
•A six sided die is labeled 1,2,3,4,5,6
•A 6 occurs only once in rolling a die
Probability =
# of ways a certain outcome can occur
# of possible outcomes
Probability =
1
6
Example 2
In a statistics class, 32 students of which 20 are females are selected to participate
in a study of eye color. It is discovered that 7 of the 32 students have blue eyes. It
is also noted that 5 out of the 20 females have blue eyes.
Males
Blue Eyes
No Blue Eyes
Total
Females
Total
Answer:
Males
Females
Total
Blue Eyes
2
5
7
No Blue Eyes
10
15
25
Total
12
20
32
1.
P(Blue eyes) = 7/32 = 0.219
2.
P(Females) = 20/32 = 0.625
3.
P(Males) = 12/32 = 0.375
4.
P(No Blue eyes) = 25/32 = 0.781
Example 4 :
In a sample of 50 people, 21 had type “O” blood, 22 had type “A”, 5 had type “B”
blood and 2 had type “AB” blood. Set up a frequency distribution and find the
following probabilities:
1. A person has type “O” blood.
2. A person has type “A” or type “B” blood.
3. A Person had neither “A” nor type “O” blood.
4. A person does not have type “AB” blood.
Example 4 :
In a sample of 50 people, 21 had type “O” blood, 22 had type “A”, 5 had type “B”
blood and 2 had type “AB” blood. Set up a frequency distribution and find the
following probabilities:
Group
Frequency
Type “O”
21
Type “ A”
22
Type “B”
5
Type “AB”
2
Total
50
Answer:
Group
Frequency
Type “O”
21
Type “ A”
22
Type “B”
5
Type “AB”
2
Total
50
1. A person has type “O” blood. P (X = “O”) = 21/50 = 0.42
2. A person has type “A” or type “B” blood. P (X = “A” or “B”) = 26/50 = 0.52
3. A Person had neither “A” nor type “O” blood.
P ( X = “B” or “ AB”) = 7/50 = 0.14
4. A person does not have type “AB” blood.
P ( X does not have type “AB”) = 48/50 = 0.96
Conditional Probability
Formula for Conditional Probability:
The probability that the second event B occurs given the first event A has occurred
can be found by dividing the probability that both occurred by the probability that
the first event has occurred.
P( B | A) 
A
P( A and B)
P( A)
B
A and B
Example 5 (Conditional Probability)
Males
Females
Total
Blue Eyes
2
5
7
No Blue Eyes
10
15
25
Total
12
20
32

What is the probability that a student selected at random is a female given that
the student has blue eyes?
P(Female | blue eyes) = 5/7 = 0.714

What is the probability that a student selected at random has blue eyes given that
the student is male?
P(Blue eyes| Male) = 2/12 = 0.167
Compound Probabilities

Events that occur in combination
• P(blue eyes and female) or in general:
P(A and B)

Events that occur as alternatives
• P(blue eyes or female) or in general:
P(A or B)
Multiplication (‘AND’) Law
Equation #1: If A and B are independent, then;
P (A and B) = P(A) x P(B)
Equation #2: If A and B are not independent i.e dependent, then;
P (A and B) = P (A | B) x P (B)
or
P (B | A) x P (A)
Test of Independent Events:
•
Two events A and B are independent if the fact that A occurs does not
affect the probability of B occurring
• Two events A and B are independent events if
P(A | B) = P (A)
or
P(B | A) = P (B)
Note :If two events are not independent, they are dependent.
Example :
In a statistics class, 32 students of which 20 are females are selected to
participate in a study of eye color. It is discovered that 7 of the 32 students have
blue eyes. It is also noted that 5 out of the 20 females have blue eyes.
Males
Females
Total
Blue Eyes
2
5
7
No Blue Eyes
10
15
25
Total
12
20
32
Example :
What is the probability of being a female and having blue eyes?
Step 1: Is having blue eyes dependent on gender?
P(Blue eyes | Female) = P (Blue eye )
5/20 ≠ 7/32
Thus having blue eyes is dependent on gender.
Step 2: Use Equation #2:
P (Female and Blue eyes) = P (Blue eyes | Female) x P (Female)
= 5/20 x 20/32 = 5/32 = 0.156
Example :
A coin is flipped and a die is rolled. Find the probability of getting a head on the coin
and a 4 on the die.
Answer:
P (head and 4) = P (head). P (4)= 1/2 * 1/6 = 1/12 = 0.083
Addition (‘OR’) Law
Equation #1: If A and B are mutually exclusive:
P (A or B) = P(A) + P(B)
P(A)
(+)
P(B)
A
B
A
B
Equation #2: If A and B are not mutually exclusive:
P(A or B) = P(A) + P(B) – {P(A and B)}
P(A)
P(B)
Note:
•
Two events are mutually exclusive if they cannot occur at the
same time (i.e. P(A and B) = 0)
A and B
Example :
In a statistics class, 32 students of which 20 are females are selected to
participate in a study of eye color. It is discovered that 7 of the 32 students have
blue eyes. It is also noted that 5 out of the 20 females have blue eyes.
Males
Females
Total
Blue Eyes
2
5
7
No Blue Eyes
10
15
25
Total
12
20
32
Example :
What is the probability a student selected at random will be a female or has blue
eyes?
Answer:
Step 1: Is having blue eyes and gender mutually exclusive?
Since a given individual can be a female and have blue eyes, thus they are not
mutually exclusive.
Step 2: Equation #2:
P(Blue eyes or Female) = P(Blue Eyes) + P(Female) – [P(Blue eyes and Female)] =
7/32 + 20/32 – 5/32 = 22/32 = 0.688
Example :
A day of the week is selected at random. Find the probability that it is a
weekend day.
Answer:
P (Saturday or Sunday) = P ( Saturday) + P (Sunday)
= 1/7 + 1/7 = 2/7 = 0.286