RandomVariables-6

RandomVariables-6 - From Urns to Coupons Coupon Collecting...

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1 From Urns to Coupons “Coupon Collecting” is classic probability problem There exist N different types of coupons Each is collected with some probability p i (1 ≤ i N ) Ask questions like: After you collect m coupons, what is probability you have k different kinds? What is probability that you have ≥ 1 of each N coupon types after you collect m coupons? You’ve seen concept (in a more practical way) N coupon types = N buckets in hash table collecting a coupon = hashing a string to a bucket Digging Deeper on Independence Recall, two events E and F are called independent if P(EF) = P(E) P(F) If E and F are independent, does that tell us anything about: P(EF | G) = P(E | G) P(F | G), where G is an arbitrary event? In general, No! Not-so Independent Dice Roll two 6-sided dice, yielding values D 1 and D 2 Let E be event: D 1 = 1 Let F be event: D 2 = 6 Let G be event: D 1 + D 2 = 7 E and F are independent P(E) = 1/6, P(F) = 1/6, P(EF) = 1/36 Now condition both E and F on G: P(E|G) = 1/6, P(F|G) = 1/6, P(EF|G) = 1/6 P(EF|G) P(E|G) P(F|G) E|G and F|G dependent Independent events can become dependent by conditioning on additional information Do CS Majors Get Less A’s? Say you are in a dorm with 100 students 10 of the students are CS majors: P(CS) = 0.1 30 of the students get straight A’s: P(A) = 0.3 3 students are CS majors who get straight A’s o P(CS, A) = 0.03 o P(CS, A) = P(CS)P(A), so CS and A are independent At faculty night, only CS majors and A students show up o So, 37 (= 10 + 30 – 3) students arrive o Of 37 students, 10 are CS P(CS | CS or A) = 10/37 = 0.27 o Appears that being CS major lowers probability of straight A’s o But, weren’t they supposed to be independent? In fact, CS and A conditionally dependent at faculty night Explaining Away Say you have a lawn It gets watered by rain or sprinklers P(rain) and P(sprinklers were on) are independent Now, you come outside and see the grass is wet o You know that the sprinklers were on
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RandomVariables-6 - From Urns to Coupons Coupon Collecting...

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