Chris Morgan, MATH G160 January 18, 2012 Lecture 4 Chapter 4.4: Independence 1

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3 Two events are independent if the occurrence of one of the events gives us NO information about whether or not the other event will occur; that is, the events have no influence on each other. In probability theory we say that two events, A and B, are independent if the probability that they both occur is equal to the product of the probabilities of the two individual events:

Independence 4 The idea of independence can be extended to more than two events. For example, A, B and C are independent if: - A and B are independent - A and C are independent and B and C are independent (pairwise independence)

Independence 5 Suppose that a man and a woman each have a pack of 52 playing cards. Each draws a card from his/her pack. Find the probability that they each draw the ace of clubs. We define the events: - A = probability that man draws ace of clubs = 1/52 - B = probability that woman draws ace of clubs = 1/52 Clearly events A and B are independent so: P(A ∩ B) = P(A)P(B) = 1/52 * 1/52 = That is, there is a very small chance that the man and the woman will both draw the ace of clubs.

Deck of cards breakdown: 52 total, 26 red and 26 black, of the red we have 13 hearts, 13 diamonds, of the black we have 13 spades and 13 clubs P(8) =P(8|H) = P(8 c |H c ) = Independence 6 HA JQK DA JQK CA JQK SA JQK

Deck of cards breakdown: 52 total, 26 red and 26 black, of the red we have 13 hearts, 13 diamonds, of the black we have 13 spades and 13 clubs These events are known as independent events. Knowing the outcome of one event (heart) does not influence the outcome of the other event (eight). Independence 7 HA JQK DA JQK CA JQK SA JQK

Independence 8 In general: If A and B are events of a sample space, where P(B)>0 (not an impossible event), A is independent of B if the occurrence of B does not affect the probability that A occurs. If P(A|B) = P(A) then A and B are independent. When A and B are independent: A and BC ; AC and B and AC and BC, will all be independent of each other. To PROVE Independence show either : or:

Deck of cards breakdown: 52 total, 26 red and 26 black, of the red we have 13 hearts, 13 diamonds, of the black we have 13 spades and 13 clubs P (Red and 10) = P(Face Card and Black) = Independence 9 HA JQK DA JQK CA JQK SA JQK

Deck of cards breakdown: 52 total, 26 red and 26 black, of the red we have 13 hearts, 13 diamonds, of the black we have 13 spades and 13 clubs P (Diamond | {2, 3, 4, 5}) = P (2 and spade | black) = Independence 10 HA JQK DA JQK CA JQK SA JQK

Deck of cards breakdown: 52 total, 26 red and 26 black, of the red we have 13 hearts, 13 diamonds, of the black we have 13 spades and 13 clubs Are the events [Ace] and [Heart] independent? How about [Red] and [10] or [Face Card] and [Black]? Independence 11 HA JQK DA JQK CA JQK SA JQK

Independence [example 11] 12 A red die and a white die are rolled: Event A = {4 on the red die} Event B = {sum of dice is odd} Find the following: a) P(A) c) P(A B) b) P(B) d) P(A)P(B)

Independence [example 12] 13 Insurance companies assume that there is a difference between gender and your likelihood of getting into an accident which is why women generally have lower insurance rates than men. We did a study to see the number of accidents that occurred according to gender. We found: 60% of the population was male 86% of the population was either male OR got into an accident 35% of the population are accident free Does this study indicate that the likelihood of one to get into an accident depends on gender?

Independence [example 13] 14 Flip a fair coin five independent times. Compute the probability of : a) HHHTT b) HTHTH c) TTHHT d) Three heads occurring in the five trials?

Independence [example 14] 15 An urn contains two red balls and four white balls. Sample successively five times at random and with replacement so that the trials are independent. Compute the probability of: a) WWRRW b) RRWWR c) Four whites in five trials?

Independence [example 15] 16 An urn contains five balls, one marked WIN and four marked LOSE. You and another player take turns selecting a ball from the urn, one at a time. The first person to select the WIN ball is the winner. If you draw first, find the probability that you will win if the sampling is done: a) Without replacement b) With replacement

Independence [example 16] 17 An urn contains 10 red ball and 10 white balls. The balls are drawn from the urn at random, one at a time. Find the probability that the fourth white ball is the sixth ball drawn if the sampling is done: a) With replacement b) Without replacement c) In the World Series the American League (red) and the National League (white) teams play until one team wins four games. Do you think this urn model could be used to describe the probabilities of a 4-, 5-, 6-, or 7-games series? If your answer is yes, would you choose sampling with or without replacement?

Independence [example 17] 18 Suppose you are fishing in a pond which contains 20 fish of which 12 are bass (B) and 8 are blue gill (G). Every time you catch a fish you record its species and release it back into the pond. If we assume the fish have no memory, then every time you catch and release a fish it is like you are starting all over again, in other the trials (species you catch) will be independent. If you decide to fish until you catch 10, find the probability of observing the following sequences of caught fish species: a) BGBGBGBGBG b) GGGGGBBBBB c) BBBBBBBBBB d) At least one blue gill e) Exactly 5 bass and 5 blue gill

Independence [example 18] 19 The Minnesota Twins win with a probability 0.6. All games are independent and the success or failure of the Twins does not depend on past successes or failures (so no streaks of any kind). You watch the next 3 games. P(W1 and W2) = P(win next 3 games) = P(win AT LEAST 1 of next 3 games) = P(win EXACTLY 1 of next 3 games) = P(1st win is the 3rd game) =

Independence [example 19] 20 Me and my roommates each have a car: Julia has a Mercedes-Benz SLK which works with probability 0.98 Alex has a Mercelago Diablo which works with probability 0.91 Chris has a P.O.S GMC Jimmy which works with probability 0.24 P(at least one car works) = P(Exactly one works) =