Probability Models. The of a chance process is the set of all possible outcomes.

Transcription

1 Prbability Mdels In Sectin 5.1, we used simulatin t imitate chance behavir. Frtunately, we dn t have t always rely n simulatins t determine the prbability f a particular utcme. Descriptins f chance behavir cntain tw parts: DEFINITION: Sample space, prbability mdel The f a chance prcess is the set f all pssible utcmes. The is a descriptin f sme chance prcess that cnsists f tw parts: a sample space S and a prbability fr each utcme. Rll the Dice (Building a prbability mdel) Many bard games invlve rlling dice. Imagine rlling tw fair, six-sided dice ne that s red and ne that s green. PROBLEM: Give a prbability mdel fr this chance prcess. Prbability Mdels Prbability mdels allw us t find the prbability f any cllectin f utcmes, which we call an. DEFINITION: Event An event is any cllectin f utcmes frm sme chance prcess. That is, an event is a subset f the sample space. Events are usually designated by capital letters, like A, B, C, and s n.

2 If A is any event, we write its prbability as P(A). In the dice-rlling example, suppse we define event A as sum is 5. There are 4 utcmes that result in a sum f 5. Since each utcme has prbability 1/36, P(A) = 4/36. Suppse event B is defined as sum is nt 5. What is P(B)? P(B) = 1 4/36 = 32/36 Let s cnsider ne mre event, which we ll call C: sum is 6. The utcmes in event C are S P(C) = 5/36. What s the prbability that we get a sum f 5 r 6, that is, P(A r C)? Basic Rules f Prbability All prbability mdels must bey the fllwing rules: The prbability f any event is a number between and. All pssible utcmes tgether must have prbabilities whse sum is. If all utcmes in the sample space are equally likely, the prbability that event A ccurs can be fund using the frmula number f utcmes crrespnding t event A P(A) ttal number f utcmes in sample space The prbability that an event des nt ccur is. If tw events have n utcmes in cmmn, the prbability that ne r the ther ccurs is the sum f their individual prbabilities. DEFINITION: Mutually exclusive (disjint) Tw events are mutually exclusive (disjint) if they have n utcmes in cmmn and s can never ccur tgether. Basic Rules f Prbability (symblic frm) Fr any event A,. If S is the sample space in a prbability mdel,. In the case f equally likely utcmes, P(A) = Cmplement Rule: P(Ac) = Additin rule fr mutually exclusive events: If A and B are mutually exclusive, P(A r B) =

3 Distance Learning (Applying prbability rules) Distance-learning curses are rapidly gaining ppularity amng cllege students. Randmly select an undergraduate student wh is taking a distance-learning curse fr credit, and recrd the student s age. Here is the prbability mdel: Age grup (yr): 18 t t t r ver Prbability: PROBLEM: (a) Shw that this is a legitimate prbability mdel. (b) Find the prbability that the chsen student is nt in the traditinal cllege age grup (18 t 23 years.) Tw-Way Tables and Prbability When we re trying t find prbabilities invlving tw events, a tw-way table can display the sample space in a way that makes prbability calculatins easier. Wh Has Pierced Ears? (Tw-way tables and prbability) Students in a cllege statistics class wanted t find ut hw cmmn it is fr yung adults t have their ears pierced. They recrded data n tw variables gender and whether the student had a pierced ear fr all 178 peple in the class. The tw-way table belw displays the data. PROBLEM: Suppse we chse a student frm the class at randm. Find the prbability that the student (a) has pierced ears. (b) is a male with pierced ears. (c) is male r has pierced ears.

4 Nte, the previus example illustrates the fact that we can t use the additin rule fr mutually exclusive events unless the events have n utcmes in cmmn. The belw explains why. General Additin Rule fr Tw Events If A and B are any tw events resulting frm sme chance prcess, then P(A r B) = CHECK YOUR UNDERSTANDING A standard deck f playing cards (with jkers remved) cnsists f 52 cards in fur suits clubs, diamnds, hearts, and spades. Each suit has 13 cards, with denminatins ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king. The jack, queen, and king are referred t as face cards. Imagine that we shuffle the deck thrughly and deal ne card. Let s define events: A: getting a face card and B: getting a heart. 1. Make a tw-way table that displays the sample space. 2. Find P(A and B) 3. Explain why P(A r B) P(A) + P(B). Then use the general additin rule t find P(A r B). Venn Diagrams and Prbability Because Venn diagrams have uses in ther branches f mathematics, sme standard vcabulary and ntatin have been develped.

5 The cmplement A C cntains exactly the utcmes that are nt in A. The events A and B are mutually exclusive (disjint) because they d nt verlap. That is, they have n utcmes in cmmn. The intersectin f events A and B (A B) is the set f all utcmes in bth events A and B. The unin f events A and B (A B) is the set f all utcmes in either event A r B. Wh Has Pierced Ears? (Understanding Venn diagrams) In the preceding example, we lked at data frm a survey n gender and ear piercings fr a large grup f cllege students. The chance prcess came frm selecting a student in the class at randm. Our events f interest were A: is male and B: has pierced ears. Here is the tw-way table that summarizes the sample space.

6 Hw wuld we cnstruct a Venn diagram that displays the infrmatin in the tw-way table? There are fur distinct regins in the Venn diagram shwn in the Venn diagram we just cnstructed. These regins crrespnd t the fur cells in the tw-way table. We can describe this crrespndence in tabular frm as fllws: With this new ntatin, we can rewrite the general additin rule in symbls as P(A U B) = Wh Reads the Paper? (Venn diagrams, tw-way tables, and prbability) In an apartment cmplex, 40% f residents read USA Tday. Only 25% read the New Yrk Times. Five percent f the residents read bth papers. Suppse we select a resident f the apartment cmplex at randm and recrd which f the tw papers the persn reads. PROBLEM: (a) Make a tw-way table that displays the sample space f this chance prcess. (b) Cnstruct a Venn diagram t represent the utcmes f this chance prcess. (c) Find the prbability that the persn reads at least ne f the tw papers. (d) Find the prbability that the persn desn t read either paper.