5.3. Independence and the Multiplication Rule

Transcription

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 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 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) E c = the complement of E, when E does not happen E c = exactly zero Often it is easier to compute P(E c ) first, and then compute P(E) as 1 P(E c )

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} E c = {no heads} = {all tails} E c 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(E c ) = 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