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Statistics 510: Notes 4 Reading: Sections 2.4-2.5 I. Propositions about Probability Function Based on Axioms (Section 2.4) C Proposition 4.1: P( E ) 1 P( E ) . C Proof: Because E E S , by Axiom 2 we have P( E E C ) P( S ) 1 . Because E and E C are mutually exclusive, it follows from Axiom 3 that P( E E C ) P ( E ) P ( E C ) . C C Thus, P( E ) P( E E ) P( E ) 1 P( E ) . Proposition 4.2: If E F (meaning that every outcome in E is contained in F ), then P( E ) P( F ) . Proof: Note that the event F may be written in the form F E (F EC ) , where E and F E C are mutually exclusive. Therefore, by Axiom 3, P( F ) P( E ) P( F E C ) . By Axiom 1, P( F E C ) 0 so that P( F ) P( E ) . 1 Furthermore, from the proof of Proposition 4.2, we have the difference rule that if E F , P( F and not E ) P( F E C ) P( F ) P( E ) . Proposition 4.3: P( E F ) P( E ) P( F ) P( E F ) . Proof: The Venn diagram suggests the statement of the proposition is true. More formally, we have from Axiom 3 that P( E ) P ( E F C ) P ( E F ) P( F ) P( E F ) P ( E C F ) P( E F ) P( E F C ) P ( E F ) P ( E C F ) From the first two equations, we have that P( E F C ) P( E ) P( E F ), P( E C F ) P( F ) P( E F ) Substituting these expressions in the expression for P ( E F ) , we conclude that P( E F ) P( E ) P( F ) P( E F ) . Note: Proposition 4.3 can be extended to provide an expression for P( E1 E2 En ) ; see Proposition 4.4, the inclusion-exclusion identity). 2 Example 1: Winthrop, a premed student, has been summarily rejected by all 126 U.S. medical schools. Desperate, he sends his transcripts and MCATs to the two least selective campuses he can think of, the two branch campuses ( X and Y ) of Swampwater Tech. Based on the success his friends have had there, he estimates that his probability of being accepted at X is 0.7, and at Y , 0.4. He also suspects that there is a 75% chance that at least one of his applications will be rejected. What is the probability that at least one of the schools will accept him? 3 II. Sample Spaces Having Equally Likely Outcomes For many experiments, it is natural to assume that all outcomes in the sample space are equally likely to occur. That is, consider an experiment whose sample space S is a finite set, say S {1, 2, , N } . Then it is often natural to assume that P({1}) P({2}) P({N }) . This implies that 1 P({i}) , i 1, , N N because P( S ) 1 by Axiom 2 and P( S ) P({1}) P({N }) by Axiom 3. For a sample space with equally likely outcomes, for any event E , number of outcomes in E P( E ) (0.1) number of outcomes in S (this follows from Axiom 3). In other words, if we assume that all outcomes of an experiment are equally likely to occur, then the probability of an event E equals the proportion of outcomes in the sample space that are contained in E . For counting the number of outcomes in E and the number of outcomes in S , the methods of Chapter 1 are useful. 4 Example 2: Your friend at Penn has a phone number that starts with 498. Suppose that the remaining four digits in your friend’s phone number are equally likely to be any sequence. What is the probability that your friend’s phone number contains seven distinct digits. 5 Example 3: A poker hand consists of 5 cards. If the cards have distinct consecutive values and are not all of the same suit, we say that the hand is a straight. For instance, a hand consisting of the five of spades, six of spades, seven of spades, eight of spades and nine of hearts is a straight? What is the probability that one is dealt a straight? 6 Example 4: A deck of 52 playing cards is well shuffled and the cards turned up one at a time until the first ace appears. Is the next card – that is, the card following the first ace – more likely to be the ace of spades or the two of clubs? 7 Example 5: A football team consists of 20 offensive and 20 defensive players. The players are to be paired in groups of 2 for the purpose of determining roommates. If the pairing is done at random, what is the probability that there are no offensive-defensive roommate pairs? What is the probability that there are 2i offensive-defensive roommate pairs, i=1,2,...,10? 8