Scribd
Upload
3 views23 pages

Conditional Probability

The document discusses conditional probability and the multiplication rules for independent and dependent events, providing examples for each. It explains how to calculate the probability of two independent events occurring in sequence and illustrates dependent events through card and ball selection scenarios. Additionally, it presents a case study involving an electrical system with four components to demonstrate the application of conditional probability in determining system reliability.

Uploaded by

husseinazzam880
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
3 views23 pages

Conditional Probability

The document discusses conditional probability and the multiplication rules for independent and dependent events, providing examples for each. It explains how to calculate the probability of two independent events occurring in sequence and illustrates dependent events through card and ball selection scenarios. Additionally, it presents a case study involving an electrical system with four components to demonstrate the application of conditional probability in determining system reliability.

Uploaded by

husseinazzam880
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 23

Conditional probability

Clara Al Kosseifi

clara.kosseifi@fty.balamand.edu.lb
Multiplication rules
The multiplication rules can be used to find the probability of two or more
events that occur in sequence.

Examples:
• If you toss a coin and then roll a die, you can find the probability of
getting a head on the coin and a 4 on the die. These 2 events are called
independent since the outcome of the first event does not affect the
probability outcome of the second event.
• Rolling a die and getting a 6 and then rolling a second die and getting a 3
• Drawing a card and getting a queen and replacing it and drawing a
second card and getting a queen
Probability of two Independent
events
To find the probability of two independent events that occur in sequence,
you must find the probability of each event occurring separately and then
multiply the answers.

Example: If a coin is tossed twice, the probability of getting two heads is


1 1 1
× = =0.25
2 2 4

This result can be verified by looking at the sample space HH, HT, TH, TT
1
then P(HH)=
4
Examples
Examples
Examples
Examples
Examples
Dependent events
Examples:
• Suppose a card is drawn from a deck and not replaced and then a second
4
card is drawn. The probability of selecting an ace on the first draw is .
52
If the card is not replaced the probability of selecting a king on the
4
second draw since there are 4 kings and 51 cards remaining
51

➪P(AK)= × =
4 4 16 4
= ~0.006.
52 51 2652 663

• Selecting a ball from an urn, not replacing it, and then selecting a second
ball
Conditional probability
Conditional probability
Examples
Conditional probability
Examples
Examples
Examples
Examples
Examples
An electrical system consists of four components as illustrated and the
reliability (probability of working) of each component is also shown.

Find the probability that


(a) the entire system works
(b) the component C does not work, given that the entire system works.
Assume that the four components work independently.
Examples
(a) the entire system works: 𝑃𝑃 𝐴𝐴 ∩ 𝐵𝐵 ∩ 𝐶𝐶 ∪ 𝐷𝐷 = 𝑃𝑃 𝐴𝐴 . 𝑃𝑃 𝐵𝐵 . 𝑃𝑃 𝐶𝐶 ∪ 𝐷𝐷
= 𝑃𝑃 𝐴𝐴 . 𝑃𝑃 𝐵𝐵 . 1 − 𝑃𝑃 𝐶𝐶 ′ ∩ 𝐷𝐷′ = 𝑃𝑃 𝐴𝐴 . 𝑃𝑃 𝐵𝐵 . 1 − 𝑃𝑃 𝐶𝐶 ′ . 𝑃𝑃 𝐷𝐷′
= 0.9 ∗ 0.9 ∗ 1 − 1 − 0.8 ∗ 1 − 0.8 = 0.7776

(b) the component C does not work, given that the entire system works:
𝐶𝐶 ′ 𝑃𝑃(𝐶𝐶 ′ ∩ 𝐴𝐴 ∩ 𝐵𝐵 ∩ 𝐶𝐶 ∪ 𝐷𝐷 ) 𝑃𝑃(𝐶𝐶𝐶 ∩ 𝐴𝐴 ∩ 𝐵𝐵 ∩ 𝐷𝐷)
𝑃𝑃 = =
𝐴𝐴 ∩ 𝐵𝐵 ∩ 𝐶𝐶 ∪ 𝐷𝐷 𝑃𝑃(𝐴𝐴 ∩ 𝐵𝐵 ∩ 𝐶𝐶 ∪ 𝐷𝐷 ) 𝑃𝑃(𝐴𝐴 ∩ 𝐵𝐵 ∩ 𝐶𝐶 ∪ 𝐷𝐷 )
𝑃𝑃 𝐶𝐶 ′ . 𝑃𝑃 𝐴𝐴 . 𝑃𝑃 𝐵𝐵 . 𝑃𝑃(𝐷𝐷) 1 − 0.8 ∗ 0.9 ∗ 0.9 ∗ 0.8
= = = 0.1667
0.7776 0.7776
Probability for at Least
Probability for at Least
Probability for at Least
Extension: Multiple events

So to prove that events are independent we need to prove that the


probability of their intersection is equal to the product of their
probabilities:

You might also like