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2-16 - Chapter 4 — Probabilities and Simulation Addition...

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Unformatted text preview: Chapter 4 — Probabilities and Simulation Addition ‘rule for two mutually exclusive events . If events A and Bare mutually exclusive, then the probability that A or Boccurs is the sum of the probabilities of each event, l P(AorB)=P(A)+ P(B). , ' This rule also works for more than two mutually exclusive events. ,, Example 9: Rolling a Die, revisited ~ l What is the probability of rolling either a 4 or a 6 on one roll of'a die, or P(4 or 6)? i more» = name): yew/c. = V?) What is the probability of not rolling either a 4 or a6? Or what is P(not 4 or 6)? P(no£,/IH org"): I'P(Hor(9>: l5‘/3:Z/5 Addition rule for events not mutually exclusive '1 If events A and Bare not mutually exclusive, then the addition rule is P(A or B) = P(A) + P(B) — P(A and B). So event (A or B) is the event that at least one of the events A or Boccurs, and event (A and B) is the event that both events A and 5 occur. @‘i Example 10: Rolling Dice ' Suppose you toss a pair of dice — one orange and one blue. Consider the events A = (the orange die shows three clots) and 'B = (the blue die shows three clots). Find the following probabilities. a. P(A) -: l/(p ' cl. P(AorB) , ' ’: l/ + 1/ l , H (a 6: " /3&; “ 54, ‘ b. P(B) -: ‘/(0 e. P(atleastone 3) _ same as (A) =— H/ga .. l | l c.PAandB _....,.:__.._ f. PnotA ‘/ 2&9 HS) G 6 31a _( ) _\ - 5 f “a ' —— ItPCA3=l /Co" /a neth< at ‘5 - _ As was seen in Example 8, a probability tree displays on each branch of the tree the probability of the next stage outcome assuming that the present stage outcome has occurred. It is especially useful when outcomes are not equally likely. Example 11: Clothing Jill has two pairs of shoes, three pairs of pants, and four blouses to choose from when picking out what to wear to work each day. Construct a tree diagram to illustrate her - B options. / P! 4%}; . \ Pi ‘ 52< PLE‘ F55. 52 Chapter 4 — probabilities and Simulation Multiplication rule for probability trees P(outcome) = product of probabilities along the appropriate branches Multiplication rule for independent events > i. For independent events A and 5, PM and B): P(A ) >< P(B). Example 12: Drawing Balls A bag contains 4 green balls (G) and 6 black balls (B). Patrick picks a ball at random from the bag, then puts it back in the bag, and then draws a second ball. Construct a probability tree, including the corresponding probabilities at each level, to illustrate the outcomes. Then use the multiplication rules to find the following probabilities. Are these events independent? 0;} G : 0.19 W/(£P‘QC€MCY1£ : l‘flJCPénolen‘é- [ 0} B 1' O.Zt—‘ LO/O (EPLKCCMCQI: Z dePenden-t 0H (2 ' 1<50-“ 6 = (9.214 |°~cl5 : 0.30 a_ P(G,G) :_ 0H0 d. P(B,G) 7: 07—") b. P(B, B) = 035cc e. P(Band G) 1 orzqmzuwug c. P(G, B) :1 01:4 f. P(BorG) +lu‘5 must hqpfien 2,1 Conditional probability of 3, given A P(BIA) 2 PM a;d(j)occur) Example 13: Deck of Cards Answer the following probabilities pertaining to a regular deck of cards. A deck of cards has four suits: clubs, diamonds, hearts, and spades. There are 13 cards in each suit: numbers 2 through 10, jack, queen, king, and ace. The jacks, queens, and kings are face cards. a. P(king) :— i : J— r c. P(king I face card) ;—_ P(‘pqtc moi AND kins) Z '3 P (-9“: «NA 1 b. P(club) V5 l ' . . . P C k‘ r: “T" :— ""‘ _ P(ki‘nsgml Club) T \i 5 2 L1 ' P he“.c Md) - - P(Club) 1 ; POM: «#clubs> C‘- /'3 53 P(club) m#151 Chapter 4 — Probabilities and Simulation General multiplication rule for two events and for three events For two events A and B . P(AandB)=P(A)xP(B|A) For three events A, B, and C PM and Band Call occur) = P(A ) x P(B |A ) x P (CIA and Boccur) Example 14: Deck of Cards, revisited Suppose a regular deck of cards is shuffled Well and that three cards are dealt one at a time to Daniel. Find the following probabilities. a. P (first two cards are clubs) d. P (ten, five, two) : Pug. .nd Q): P(%)-P(‘<32l%) 1 it .11.. ‘4 2 0.00048 - 2 '— - 'B/EflJZ/E).10.55,:l 5 5! so b. P (all 3 cards are clubs) e. P (jack, queen, king) L:- o .L-Z— il- : O - (d) r- ‘52 51’ 50 ‘0‘?) . - frame as ~ o.oootfa c. P (all 3 cards are tens) f. P (spade, heart, diamond) , ii 3 ,2; _ ' 52'5'50'0‘000’8 12.51.920.017 £32 BI 50 Definition of Independent Events Events A and Bare independent if P (3 M) = P (3) Example 15: Skipping Class Melanie likes to skip class. Suppose the probability that Melanie attends class on Tuesday is P(attending class Tuesday) = 0.25 and the probability that she attends class on Wednesday is P(attending class Wednesday) = 0.25. Also, the probability that she attends class both days is P(attending class both days) = 0.20. Are the events A = attends class on Tuesday and B = attends class on Wednesday independent? The two fundamental rules for probabilities are 0 S P (outcome) 5 1, and Z P (outcome) = 1 all outcomes in 5 Also, the probability that any particular event A of a random experiment will occur is the sum of all probabilities of the outcomes in the event, or P (A ) = Z P (outcome) all outcomes in A ,54 ...
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