1 / 17

5.3

5.3. Independence and the Multiplication Rule. 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 )

hedy
Download Presentation

5.3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related