Mathematical Foundations of Computer Science Lecture Outline October 9, 2018

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1 Mathematical Foundations of Computer Science Lecture Outline October 9, 2018 Eample. in 4? When three dice are rolled what is the probability that one of the dice results Let F i, i {1, 2, 3} be the event that the ith dice results in a 4. We are interested in Pr[F 1 F 2 F 3 ]. By inclusion-eclusion formula we have Pr[F 1 F 2 F 3 ] = Pr[F 1 ]Pr[F 2 ]Pr[F 3 ] Pr[F 1 F 2 ] Pr[F 1 F 3 ] Pr[F 2 F 3 ]Pr[F 1 F 2 F 3 ] Since the events F 1, F 2, F 3 are mutually independent we can rewrite the above epression as Pr[F F 2 F 3 ] = Pr[F 1 ] Pr[F 2 ] Pr[F 3 ] Pr[F 1 ] Pr[F 2 ] Pr[F 1] Pr[F 3 ] Pr[F 2 ] Pr[F 3 ] Pr[F 1 ] Pr[F 2 ] Pr[F 3 ] = ( = ) ( ) ( ) ( ) 6 An easier way to solve this is as follows. Let F i be the complement of event F i, i = 1, 2, 3. Pr[F 1 F 2 F 3 ] = 1 Pr[F 1 F 2 F 3 ] = 1 (5/6) 3 = Eample. turn up? A coin is tossed 10 times. What is the probability that eight or more heads Let E i denote the event that eactly i heads turn up. We are interested in Pr[E 8 E 9 E 10 ]. Since the events E i are disjoint, we have Pr[E 8 E 9 E 10 ] = Pr[E 8 ] Pr[E 9 ] Pr[E 10 ] Note that Pr[E i ] = ( ) 10 i /2 10. ence, we have Pr[E 8 E 9 E 10 ] = (( ) 10 8 ( ) 10 9 ( )) 10 = Eample. (Birthday Parado) Suppose there are k people in a room and n days in a year. We are interested in the probability that there are at least two people in the room with the same birthday. What is the smallest value of k for which this probability is at least? Assume that it is equally likely for a person to be born on any of the n days of the year.

2 2 Lecture Outline October 9, 2018 Let B be the event that at least two people in the room have the same birthday. We are interested in. = 1 = P (n, k) 1 n k For n = 365, the smallest value of k for which the RS is at least is 23. If k = 40 then = 0.89, and if k = 60 then = his means that if there are 60 people then it is almost certain that there eists two among them sharing the same birthday. o illustrate how good our model is, consider the set of presidents of the United States of America. hrough Bill Clinton, 41 people belong to this set. he chances of two of them sharing the same birthday is at least 89%. Indeed, James Polk (11th president) and Warren arding (29th president) are both born on Nov. 2. Conditional Probability We now introduce a very important concept of conditional probability. Conditional probability allows us to calculate the probability of an event when some partial information about the result of an eperiment is known. As we shall see conditional probability is often a convenient way to calculate probabilities even when no information about the result of an eperiment is available. Suppose we want to calculate the probability of event A given that event B has already occured. We denote this by Pr[A B] (read as the probability of A given B ). Since we know that event B has occured our sample space reduces to the outcomes in B. Is this a valid probability space? No, because the sum of probabilities of the outcomes in B is less than 1. ow do we change the probabilities so that this is a valid probability distribution while making sure that the relative probabilities of outcomes in B do not change? We do 1 this by scaling the probability of all sample points in B by. hus for each sample point ω B, Pr[ω B] = Pr[ω] o calculate Pr[A B] we just sum up the probabilities of sample points in A B. hus we get Pr[A B] = Pr[ω B] = Pr[ω] Pr[A B] = ω A B ω A B In order to avoid division by 0, we only define Pr[A B] when > 0. Conditional probabilities can sometimes get tricky. o avoid pitfalls, it is best to use the above mathematical definition of conditional probability. Note that the R..S. of the above equation are unconditional probabilities. Eample. Suppose we flip two fair coins. What is the probability that both tosses give heads given that one of the flips results in heads? What is the probability that both tosses give heads given that the first coin results in heads?

3 October 9, 2018 Lecture Outline 3 (1/4) (1/4) (1/4) (1/4) Coin 1 Coin 2 outcomes(prob) event B event C event A&B event A&C Figure 1: ree diagram for the eperiment in Eample 1. Eample. Suppose we flip two fair coins. What is the probability that both tosses give heads given that one of the flips results in heads? What is the probability that both tosses give heads given that the first coin results in heads? We consider the following events to answer the question. A: event that both flips give heads. B: event that one of the flips gives heads. C: event that the first coin flip gives heads. Let s first calculate Pr[A B]. Pr[A B] = Pr[A B] = Pr[A] = 1/4 3/4 = 1 3. Similarly we can calculate Pr[A C] as follows. Pr[A C] = Pr[A C] Pr[C] = Pr[A] Pr[C] = 1/4 = 1 2. he above analysis also follows from the tree diagram in Figure 1. he Multiplication Rule. For any two events A 1 and A 2 we have Pr[A 1 A 2 ] = Pr[A 1 ] Pr[A 2 A 1 ] he above formula follows from the definition of Pr[A 2 A 1 ]. his formula can be generalized to n events. We state the generalization without proof. Pr[A 1 A 2 A n ] = Pr[A 1 ] Pr[A 2 A 1 ] Pr[A 3 A 1 A 2 ] Pr[A n A 1 A 2 A 3 A n 1 ]

4 4 Lecture Outline October 9, 2018 Eample. he probability that a new car battery functions for over 10,000 miles is 0.8, the probability that it functions for over 20,000 miles is 0.4, and the probability that it functions for over 30,000 miles is 0.1. If a new car battery is still working after 10,000 miles, what is the probability that (i) its total life will eceed 20,000 miles, (ii) its additional life will eceed 20,000 miles? We will consider the following events to answer the question. L 10 : event that the battery lasts for more than 10K miles. L 20 : event that the battery lasts for more than 20K miles. L 30 : event that the battery lasts for more than 30K miles. We know that Pr[L 10 ] = 0.8, Pr[L 20 ] = 0.4 and Pr[L 30 ] = 0.1. We are interested in calculating Pr[L 20 L 10 ] and Pr[L 30 L 10 ]. Pr[L 20 L 10 ] = Pr[L 20 L 10 ] Pr[L 10 ] = Pr[L 20] Pr[L 10 L 20 ] 0.8 = = 1 2 By doing similar calculations it is easy to verify that Pr[L 30 L 10 ] = 1 8. Eample. An urn initially contains 5 white balls and 7 black balls. Each time a ball is selected, its color is noted and it is replaced in the urn along with two other balls of the same color. Compute the probability that the first two balls selected are black and the net two white. We will consider the following events to answer the question. B 1 : event that the first ball chosen is black. B 2 : event that the second ball chosen is black. W 3 : event that the third ball chosen is white. W 4 : event that the fourth ball chosen is white. We are interested in calculating Pr[B 1 B 2 W 3 W 4 ]. Using the Multiplication rule we get, Pr[B 1 B 2 W 3 W 4 ] = Pr[B 1 ] Pr[B 2 B 1 ] Pr[W 3 B 1 B 2 ] Pr[W 4 B 1 B 2 W 3 ] = =

5 October 9, 2018 Lecture Outline 5 he otal Probability heorem. Consider events E and F. Consider a sample point ω E. Observe that ω belongs to either F or F. hus, the set E is a disjoint union of two sets: E F and E F. ence we get Pr[E] = Pr[E F ] Pr[E F ] = Pr[F ] Pr[E F ] Pr[F ] Pr[E F ] In general, if A 1, A 2,..., A n form a partition of the sample space and if i, Pr[A i ] > 0, then for any event B in the same probability space, we have = n Pr[A i B] = i=1 n Pr[A i ] Pr[B A i ] i=1