5.3

1. 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 MultiplicationRule 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 • 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 • 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