Algebra of events probability ppt

Probability

Definition:

Probability: the chance an event will happen.

# 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 probability of zero indicate impossibility
A probability of one indicates certainty
*

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

Example 1:

What is the probability of rolling a 6 on a six sided die?

A six sided die is labeled 1,2,3,4,5,6
A 6 occurs only once in rolling a die
Know:

# of ways a certain outcome can occur

# of possible outcomes

Probability =

1

6

Probability =

Simple Probability

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 Females Total
Blue Eyes
No Blue Eyes
Total
Answer:

What is the probability of picking a person at random from this group with blue eyes?
What is the probability that a student selected at random is a female?
What is the probability that a student selected at random is a male?
What is the probability that a student selected at random does not have blue eyes?
Males Females Total
Blue Eyes 2 5 7
No Blue Eyes 10 15 25
Total 12 20 32
Answer:

P(Blue eyes) = 7/32 = 0.219
P(Females) = 20/32 = 0.625
P(Males) = 12/32 = 0.375
P(No Blue eyes) = 25/32 = 0.781
Males Females Total
Blue Eyes 2 5 7
No Blue Eyes 10 15 25
Total 12 20 32
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:

A person has type “O” blood.
A person has type “A” or type “B” blood.
A Person had neither “A” nor type “O” blood.
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:

A person has type “O” blood. P (X = “O”) = 21/50 = 0.42
A person has type “A” or type “B” blood. P (X = “A” or “B”) = 26/50 = 0.52
A Person had neither “A” nor type “O” blood.
P ( X = “B” or “ AB”) = 7/50 = 0.14

A person does not have type “AB” blood.
P ( X does not have type “AB”) = 48/50 = 0.96

Group Frequency
Type “O” 21
Type “ A” 22
Type “B” 5
Type “AB” 2
Total 50
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.

A and B

*

Example 5 (Conditional Probability)

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

Males Females Total
Blue Eyes 2 5 7
No Blue Eyes 10 15 25
Total 12 20 32
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)

Equation #2: If A and B are not mutually exclusive:

P(A or B) = P(A) + P(B) – {P(A and B)}

Note:

Two events are mutually exclusive if they cannot occur at the same time (i.e. P(A and B) = 0)
A

B

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

A probability of zero indicate impossibility
A probability of one indicates certainty
*

*

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PAandB

PBA

PA

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