1. Independence and the Multiplication Rule
5.3
Independence and the Multiplication Rule

2. Multiplication Rule
The Addition Rule shows how to compute “or” probabilities
P(E or F)
under certain conditions
The Multiplication Rule shows how to compute “and” probabilities
P(E and F)
also under certain (different) conditions

3. Independence
The “disjoint” concept corresponds to “or” and the Addition Rule … disjoint events and adding probabilities
The concept of independence corresponds to “and” and the Multiplication Rule … independent events and multiplying probabilities
Basically, events E and F are independent if they do not affect each other

4. Independence Definition of independence
Events E and F are independent if the occurrence of E in a probability experiment does not affect the probability of event F
Other ways of saying the same thing
Knowing E does not give any additional information about F
Knowing F does not give any additional information about E
E and F are totally unrelated

5. Examples Examples of independence
Flipping a coin and getting a “tail” (event E) and choosing a card and getting the “seven of clubs” (event F)
Choosing one student at random from University A (event E) and choosing another student at random from University B (event F)

6. Dependent
If the two events are not independent, then they are said to be dependent
Dependent does not mean that they completely rely on each other … it just means that they are not independent of each other
Dependent means that there is some kind of relationship between E and F – even if it is just a very small relationship

7. Examples Examples of dependence
Whether Jack has brought an umbrella (event E) and whether his roommate Joe has brought an umbrella (event F)
Choosing a card and having it be a red card (event E) and having it be a heart (event F)
The number of people at a party (event E) and the noise level at the party (event F)

8. Disjoint vs. Independent
What’s the difference between disjoint events and independent events?
Disjoint events can never be independent
Consider two events E and F that are disjoint
Let’s say that event E has occurred
Then we know that event F cannot have occurred
Knowing information about event E has told us much information about event F
Thus E and F are not independent
Example…Let E be rolling an odd number on a die…Let F be rolling an even number.
NOT INDEPENDENT!

9. Multiplication Rule
The Multiplication Rule for independent events states that
P(E and F) = P(E) • P(F)
Thus we can find P(E and F) if we know P(E) and P(F)

10. More than 2 events…
This is also true for more than two independent events
If E, F, G, … are all independent (none of them have any effects on any other), then
P(E and F and G and …)
= P(E) • P(F) • P(G) • …

11. Example Example P(E and F) P(E and F) = 1/8
E is the event “draw a card and get a diamond”
F is the event “toss a coin and get a head”
E and F are independent
P(E and F)
We first draw a card … with probability 1/4 we get a diamond
When we toss a coin, half of the time we will then get a head, or half of the 1/4 probability, or 1/8 altogether
P(E and F) = 1/8

12. Example Another example P(E and F)
E is the event “draw a card and get a diamond”
Replace the card into the deck
F is the event “draw a second card and get a spade”
E and F are independent
P(E and F)
P(E and F) = P(E) • P(F) = 1/4 • 1/4 = 1/16

13. Not Independent The previous example slightly modified
E is the event “draw a card and get a diamond”
Do not replace the card into the deck
F is the event “draw a second card and get a spade”
E and F are not independent
Why aren’t E and F independent?
After we draw a diamond, then 13 out of the remaining 51 cards are spades … so knowing that we took a diamond out of the deck changes the probability for drawing a spade
Or…Taking one card out changed the probability.

14. At Least There are probability problems which are stated:
What is the probability that "at least" …
For example
At least 1 means 1 or 2 or 3 or 4 or …
At least 5 means 5 or 6 or 7 or 8 or …
These calculations can be very long and tedious
The probability of at least 1 = the probability of 1 + the probability of 2 + the probability of 3 + the probability of 4 + …

15. Complement Rule There is a much quicker way using the Complement Rule
Assume that we are counting something
E = “at least one” and we wish to compute P(E)
Ec = the complement of E, when E does not happen
Ec = “exactly zero”
Often it is easier to compute P(Ec) first, and then compute P(E) as 1 – P(Ec)

16. Complement Rule Example
We flip a coin 5 times … what is the probability that we get at least 1 head?
E = {at least one head}
Ec = {no heads} = {all tails}
Ec consists of 5 events … tails on the first flip, tails on the second flip, … tails on the fifth flip
These 5 events are independent
P(Ec) = 1/2 • 1/2 • 1/2 • 1/2 • 1/2 = 1/32
Thus P(E) = 1 – 1/32 = 31/32

17. Summary
The Multiplication Rule applies to independent events, the probabilities are multiplied to calculate an “and” probability
Probabilities obey many different rules
Probabilities must be between 0 and 1
The sum of the probabilities for all the outcomes must be 1
The Complement Rule
The Addition Rule (and the General Addition Rule)
The Multiplication Rule