Probability

1. Probability Prof. Richard Beigel Math C067 September 27, 2006

2. Experiments • An experiment is a process that does may not always give the same result. • Performing an experiment once is called a trial. • The result of a trial is called its outcome.

3. Probability spaces • Sample point = outcome • Event = a set of outcomes • Sample space (S) = the set of all possible outcomes (S is analogous to the universal set U from the set-theory lectures) • Disjoint events are called mutually exclusive

4. Probabilities • If x is a sample point (outcome), • The probability of x is called p(x) • 0  p(x)  1 • If A is an event then • p(A) = the sum of the probabilities of all elements of A • 0  p(A)  1 • p({}) = 0 • p(S) = 1

5. Single Fair Coin Flip • S = {H,T} • p(H) = ½ • p(T) = ½

6. Single Fair 6-Sided Die Roll • S = {1,2,3,4,5,6} • p(1) = 1/6 • p(2) = 1/6 • p(3) = 1/6 • p(4) = 1/6 • p(5) = 1/6 • p(6) = 1/6

7. Soccer game • S = {Win,Lose,Tie} • p(Win) = ? • p(Lose) = ? • p(Tie) = ?

8. Equiprobable Outcomes If all outcomes are equally likely (as with a fair die or a fair coin) then • p(x) = 1/|S| • p(A) = |A|/|S| Outcomes are not always equally likely, so use these formulas with caution.

9. Single Fair 6-Sided Die Roll A single fair 6-sided die is rolled. • Let A be the event that an odd number is rolled. • A = {x  S : x is odd} = {1,3,5} • p(A) = |A|/|S| = 3/6 = ½

10. Single Fair 6-Sided Die Roll A single fair 6-sided die is rolled. • Let B be the event that a number greater than 4 is rolled. • B = {x  S : x > 4} = {5,6} • p(B) = |B|/|S| = 2/6 = 1/3

11. Single Fair 6-Sided Die Roll A single fair 6-sided die is rolled. • A  B is the event that an odd number greater than 4 is rolled. • A  B = {x  S : x is odd and x > 4} = {5} • p(A  B) = |A  B|/|S| = 1/6

12. Single Fair 6-Sided Die Roll A single fair 6-sided die is rolled. • A  B is the event that a number that is odd or greater than 4 is rolled. • A  B = {x  S : x is odd or x > 4} = {1,3,5,6} • p(A  B) = |A  B|/|S| = 4/6 = 2/3

13. Probability of Union • p(A  B) =? p(A) + p(B) • Let A = {1,3,5} 1/2 • Let B = {5,6} +1/3 • A  B = {1,3,5,6} 2/3

14. Probability of Union • p(A  B) = p(A) + p(B)  p(A  B) • Let A = {1,3,5} 1/2 • Let B = {5,6} +1/3 • A  B = {5} 1/6 • A  B = {1,3,5,6} =2/3

15. Mutually Exclusive Events • If A and B are mutually exclusive events, i.e., disjoint sets then p(A  B) = p(A) + p(B) • Why? • Because A  B = {}, • p(A  B) = p(A) + p(B)  p(A  B) • = p(A) + p(B)  p({}) • = p(A) + p(B)  0 • = p(A) + p(B)

16. Complement • A and Ac are disjoint, so • p(A  Ac) = p(A) + p(Ac) • p(S) = p(A) + p(Ac) • 1 = p(A) + p(Ac) • 1  p(A) = p(Ac) • p(Ac) = 1  p(A) • Also, p(A) = 1  p(Ac)

17. Single Fair 6-Sided Die Roll A single fair 6-sided die is rolled. • Let A be the event that a 6 is rolled • A = {6} • Ac = S  {6} = {1,2,3,4,5,6} – {6} = {1,2,3,4,5} • p(A) = 1/6 • P(Ac) = 1 – 1/6 = 5/6

18. Rolling Two Dice • Sample space = the set of all ordered pairs of die rolls • = {(x,y) : 1  x  6 and 1  y  6} • = {1,2,3,4,5,6}  {1,2,3,4,5,6} • = {1,2,3,4,5,6}2 • To save some writing we will write xy instead of (x,y)

19. {1,2,3,4,5,6}2

21. (Cartesian) Product of Two Sets • A  B = {(a,b) : a  A and b  B} • Let A = {egg roll, soup} • Let B = {lo mein, chow mein, egg fu yung} • A  B = {(egg roll,lo mein), (egg roll, chow mein), (egg roll,egg fu yung), (soup,lo mein), (soup,chow mein), (soup,egg fu yung)}

22. Rolling Two Dice

23. Probability of A  B • Outcomes must be equiprobable • P(A  B) = p(A)  p(B) • Let A = the event of rolling one die and getting a 6. p(A) = 1/6 • Then Ac is the event of rolling one die and not getting a 6. p(Ac) = 1 – p(A) = 5/6 • Ac Ac is the event of rolling two dice and not getting a 6 on either roll • p(Ac Ac) = p(Ac)  p(Ac) = (5/6)  (5/6) = 25/36

24. Probability of A  B • Then Ac is the event of rolling one die and not getting a 6. p(Ac) = 1 – p(A) = 5/6 • Ac Ac is the event of rolling two dice and not getting a 6 on either roll • p(Ac Ac) = p(Ac)  p(Ac) = (5/6)  (5/6) = 25/36 • (Ac Ac)cis the event of rolling two dice and getting a 6 on at least one roll • p((Ac Ac)c) = 1 – 25/36 = 11/36

25. Probability of A  B • Then Ac is the event of rolling one die and not getting a 6. p(Ac) = 1 – p(A) = 5/6 • AcAcAc = (Ac)3 is the event of rolling three dice and not getting a 6 on any of the rolls • p((Ac)3) = (p(Ac))3 = (5/6)3= 125/216 • (AcAcAc)cis the event of rolling three dice and getting a 6 on at least one roll • p((AcAcAc)c) = 1 – 125/216 = 91/216  0.421

26. Probability of A  B • Suppose that we roll two dice. • What is the probability that we get two 6s? • Let A be the event of getting a 6 when we roll one die • P(A  A) = p(A)  p(A) = (1/6)(1/6) = 1/36

27. 4 the hard way • Suppose that we roll two dice. • What is the probability that we get two 2s? • Let A be the event of getting a 2 when we roll one die • P(A  A) = p(A)  p(A) = (1/6)(1/6) = 1/36

28. Probability of A  B • Suppose that we roll two dice. • What is the probability that we get a 1 on the first die and a 3 on the second die? • Let A be the event of getting a 1 when we roll one die • Let B be the event of getting a 3 when we roll one die • P(A  B) = p(A)  p(B) = (1/6)(1/6) = 1/36 • In fact each particular outcome has probability 1/36

29. 4 the easy way • Suppose that we roll two dice. • What is the probability that one of the rolls is a 1 and the other is a 3? • The event in question consists of two outcomes. Let A = {(1,3),(3,1)} • The sample space S = {1,2,3,4,5,6}2 • p(A) = |A|/|S| = 2/36 = 1/18

30. Probability of A  B • Suppose that we roll two dice. • What is the probability that the sum of the rolls is 7? • Let A = {(x,y) : x+y = 7} = {(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)} • The sample space S = {1,2,3,4,5,6}2 • p(A) = |A|/|S| = 6/36 = 1/6

31. Probability of A  B • Suppose that we roll two dice. • What is the probability that the sum of the rolls is 4? • Let A = {(x,y) : x+y = 4} = {(1,3),(2,2),(3,1)} • The sample space S = {1,2,3,4,5,6}2 • p(A) = |A|/|S| = 3/36 = 1/12