Probability. Probability is a measure of a chance of a random event. Some definitions. A random experiment: An outcome of an experiment which cannot be predicted in advance. Sample space: Set of all possible outcomes of an event. It is designated by S . n(S ) is the number of outcomes in the sample space S . Event: A subset of S is an event space E . Example: S {1,2,3,4,5,6} and E {2,4,6} E S A simple event: A single point in a sample space. e.g. getting a 2 on a dice-throw is a simple event. A compound event: An event subset E with more than 1 simple event in it. The event of occurring an even number on a throw of dice is a compound event since E {2,4,6} Equally likely event: When one outcome is not preferred over another rationally. Exhaustive events: When all possible outcomes of an event are considered. Mutually exclusive events: The situation where two or more events cannot occur simultaneously. Independent or mutually independent events: Two sets of event E and E' are independent if the occurrence (or nonoccurrence) of E has no effect on the occurrence (or nonoccurrence) of E' , and vice versa. 1. Probability of an event E p ( E ) n( E ) n( S ) 2. Probability of a E c 1 p( E ) . The E c event space is complement to E. One could write this to be ~ E as well. 3. If two events A and B are mutually exclusive, A B , Null set Therefore, p ( A B ) 0 . 4. For an event E , 0 p( E ) 1. p( E ) 0 implies an improbable event p( E ) 1 implies a certain or sure-event. 5. p( E ) p( E c ) p( E ) p(~ E ) 1 6. For A, B S , A B is an event space where A or B or both occur (or at least one of A and one of B occurs). A B is an event space where both A and B must occur. Since n( A B) n( A) n( B) n( A B) we get p( A B) p( A B) p( A) p( B) p( A B) 7. We also designate p( A B) p( AB ) 8. Note that for mutually exclusive events A, B, C etc. n( A B C ) n( A) n( B) n(C ) Therefore, p( A B C ) p( A B C ) p( A) p( B) p(C ) 9. Examples. Problems. 1. A number is selected from 1 to 50. Find the probability that is a prime number. Here, S {1,2,..,50} and n( S ) 50 E {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47} and n( E ) 15 . Therefore, p( E ) 15 / 50 3 /10 2. Two dice are thrown simultaneously. What are the probabilities of getting: a. two even numbers b. getting an even number as the sum of both throws c. getting a multiple of ‘3’ as a sum The sample space S {(1,1), (1,2),...(6,6)} with n( S ) 36 . For (a), E {(2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2)...(6,6)} with n( E ) 9 . Therefore, p( E ) 1/ 4 For (b), n( E ) 18 . For first throw could be any number between 1 and 6, and given the first number, there is only three number for the second throw which would make the total even. Therefore, p(E ) =0.5 For (c), the sum could be 3, 6, 9, and 12. The event space for this would be: E {(1,2), (2,1), (3,3), (1,5), (5,1), (3,6), (6,3), (6,6)} with n( E ) 8 . Therefore, p( E ) 8 / 36 2 / 9 3. What is the probability that a leap year selected at random would contain 53 Sundays? A leap year would have 366 days, i.e. 52 weeks (52 Sundays) plus two additional days. These two consecutive days must be one of the following: {(S,M), (M,T),(T,W),(W,T),(T,F),(F,S),(S,S)} i.e. n( S ) 7 . Of these, only two pairs offer a Sunday. Therefore, n( E ) 2 . Therefore, p(53 _ Sundays) 2 / 7 4. A bag contains 5 red balls, 6 while balls and 7 black balls. What is the probability of drawing 3 balls such that (a) one draws 1 red and two whites, (b) no red. 18 5 For (a) and (b), n(S ) .For (a), n( E | 1red _ ball) and 3 1 6 5 6 n( E | 2white _ balls) . Therefore, n( E ) 75 . 2 1 2 18 Therefore, p( E ) 75 / 75 / 816 3 For (b), 3 balls would be chosen from white and black balls only. 13 Therefore, n(E ) . Therefore, p( E ) 286 / 816 143 / 408 3 5. A box contains 12 light bulbs of which 4 are defective. Three bulbs are chosen randomly. What is the probability that a. all the three bulbs are defective b. at least 2 of the bulbs are chosen defective c. at most 2 of the bulbs chosen are defective. 12 The 3 bulbs could be drawn from a set of 12 bulbs in 220 ways. 3 4 For (a), all three bulbs are defective. They could be picked up in ways 3 or 4 ways. Therefore, n( E ) 4 / 220 1/ 55 For (b), “at least” two bulbs defective means two or more defectives. Therefore, we have to find p(two defectives) + p(three defectives). 4 8 2 1 Now, p (two _ defectives ) 48 / 220 12 / 55 220 And p(three _ defectives ) 1/ 55 . Therefore, p( at least 2 defectives)=13/55. Similarly, for (c), “at most” two defectives implies events with 1 defective 4 8 1 2 + 2 defectives + 0 defective bulb. p (one _ defective ) 28 / 55 . 220 8 3 p(0 defective bulb)= 14 / 55 . Thus, p(at most two defective) = 54/55. 220 Conditional probability. Probability of an event A given that an event B is occurred is written as p( A | B) . This is the conditional probability of the event A given B and is given by p( A | B) p( A B) p( B) S B A p( A B) n( A B ) n( S ) Therefore, p( A | B) and p ( B) n( B ) n( S ) n( A B ) n( B ) Example. 1. A box contains 12 bulbs, 4 of which are defectives. 3 bulbs are drawn at random from the lot. Find the probability that all three are nondefectives. p 8 7 6 . . 12 11 10 Example. 2. For a biased coin, assume p( H ) 2 / 3 and p(T ) 1/ 3 . If heads appears then a number is selected at random from the set {1, 2, …9}. If tails appears then a number is selected at random form the set of {1,2, 3, 4, 5}. Find the probability that an even number appears. p(even _ number) p(even _ number | H ) p( H ) p(even _ number | T ) p(T ) Now, p(even _ number | H ) 4 / 9 p(even _ number | T ) 2 / 5 Therefore, p(even _ number) 4 2 2 1 9 3 5 3 Bayes’ theorem: Given the definition of conditional probability p( A | B) p( A B) p( B) We can write p( A B) p( A | B) p( B) p( B | A) p( A) Therefore, p( A | B) p( B | A) p( A) p( B) This is Bayes’ theorem. If p(A) = prior probability of event A before it is associated with B, and p( A | B) posterior probability of A given B, then Bayes’ theorem tells us how to change Prior Posterior Posterior = Likelihood function prior p( A) p( A) 1 p( A) p(~ A) We can express the posterior on ~ A via Another way to look at it. Let O( A) p(~ A | B) p( B |~ A) p(~ A) p( B) From these two, we get p( A | B) P( B | A) P( A) p(~ A | B) p( B |~ A) p(~ A) This is O( A | B) p( B | A) O( A) p( B |~ A) If we define ln O( A) ev( A) = evidence(A), we get the Bayes’ theorem in an evidence form (useful in court) p ( B | A) ev( A | B) ev( A) ln p ( B |~ A )