Probability Examples Number of times until first success

Probability Examples

Number of times until first success • Throw a fair die – How many times, on average, do you need to throw the die until you see a 1? – P(getting 1 at each throw) = 1/6 – Answer: 1/(1/6) = 6 Page 2

Number of times until first success • Throw a fair die – How many times, on average, do you need to throw the die until you see a 2 and 3 in that order? – Expected number of throws to see 2: 6 – After that, expected number of throws to see 3: 6 – Answer: 6+6 = 12 Page 3

Number of times until first success • Throw a fair die – How many times, on average, do you need to throw the die until you see a 2 and 3 , where the order does not matter? – P( get 2 or 3 in one throw ) = 1/3 – Expected number throws until you see one of 2 or 3: 3 – After that, P( get the other number in each throw) = 1/6 – Expected number of throws until you see the other number: 6 – Answer: 3+6 = 9 Page 4

Old Exam Question 1 P( at least one copy of T 1 in 20 weeks) = 1 – P( no copy of T 1 in 20 weeks) = 1 – P ( no T 1 in week 1 AND no T 1 in week 2 AND …) = 1 - P(no T 1 in week 1) P(no T 1 in week 2) … = Page 5

• Can we do this? P(all 10 types of toys in 20 weeks) = P(copy of T 1 in 20 weeks AND copy of T 2 in 20 weeks AND …) = P(copy of T 1 in 20 weeks) P(copy of T 2 in 20 weeks)… NO, events not independent • Correct way P(all 10 types of toys in 20 weeks) = 1 – P(A) Use inclusion-exclusion to calculate P(A) Page 7

Old Exam Question 2 Page 9