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6.2 – Probability

6.2 – Probability. Probability. The idea of probability is Empirical , that is probability is base on observation not theory. (We must observe many trials to find a probability). The big idea is this:

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6.2 – Probability

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  1. 6.2 – Probability

  2. Probability • The idea of probability is Empirical, that is probability is base on observation not theory. (We must observe many trials to find a probability). • The big idea is this: Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run.

  3. Probability– The proportion of times a random phenomenon will occur in a very long series of repetitions. (The long term relative frequency). • The theory of probability originated in the study of games of chance.

  4. Probability Models • Sample space (S) – The set of all possible outcomes. • Event – Any outcome or set of outcomes. (a subset of the sample space) • Probability Model – A mathematical description of the random phenomenon with 2 parts: 1. A sample space (S) 2. A way of assigning probabilities to events.

  5. Sample Space Simple sample space: • When we toss a coin, how many possible outcomes are there? • What is the sample space? We write S = {H, T}

  6. Sample Space Complex sample space: • We draw a random sample of 50,000 US households of the 113 million households in the country. What is the sample space? • The sample space contains all possible choices of 50,000 households of the 113 million households in the country. • In this case S is very large.

  7. Technique #1: Tree Diagram

  8. The Fundamental Principle of Counting • When flipping a coin 4 times there are 2 outcomes, flipping 4 times: (2 x 2 x 2 x 2 = 16) • When flipping a coin and rolling a die there are 2 outcomes and 6 outcomes: (2 x 6 = 12)

  9. More about sampling • Sampling with replacement: • You put ten slips of paper into a hat with the digits (0 – 9). You draw out a number, record the digit, then put the number back in the hat before drawing another number. • Sampling without replacement: • You put ten slips of paper into a hat with the digits (0 – 9). You draw out a number, record the digit, then draw another number without putting the first number back. • The random digit table is an example of sampling with replacement.

  10. 10 x 10 x 10 = 1000 • How many 3 digit numbers can you make if you sample with replacement? • How many 3 digit number can you make if you sample without replacement? 10 x 9 x 8 = 720

  11. Probability Rules • If A is an event, then 0 ≤ P(A) ≤ 1 • Any probability is a number between 0 and 1 • If S is the sample space in a probability model, then P(S) = 1 • The sum of the probabilities of all possible outcomes must equal 1

  12. Probability Rules • Two events A and B are disjoint (mutually exclusive) if they have no outcomes in common and therefore can never occur simultaneously. If A and B are disjoint, we apply the addition rule: P(A or B) = P(A) + P(B)

  13. Disjoint Events: P(heads or tails)? 1/2 + 1/2 =1 P(1 or 6) when rolling a die? 1/6 + 1/6 = 1/3

  14. Probability Rules The complementof any event A is the event that A does not occur, written Ac The complement rule states that the probability that an even does not occur is 1 minus the probability that the event does occur. P(Ac) = 1 – P(A)

  15. Equally likely outcomes Which of the following have equally likely outcomes? • Coin • A single die • Sum of faces of 2 dice • Digits 0 – 9 • Single digits in the numbers 0 – 100 • Deck of playing cards

  16. General Addition Rules • If A and B are disjoint, P(A or B) = P(A) + P(B). What if there are more than 2 events, or if the events are not disjoint? • If there are more than 2 disjoint events, the Addition Rule for Disjoint Events still holds: • P(A or B or C) = P(A) + P(B) + P(C) • NOTE: This rule extends to any number of disjoint events.

  17. General Addition Rules • What if events A and B are not disjoint? • They can occur simultaneously. • The Union of any collection of events is the event that at least one of the collection occurs.

  18. General Addition Rules • General Addition Rule for Unions of Two Events: • For any two events A and B: P(A or B) = P(A) + P(B) – P(A and B) or P(A U B) = P(A) + P(B) – P(A ∩ B) • If A and B are disjoint, what does P(A ∩ B) equal?

  19. General Addition Rules Deborah and Matt are waiting to find out who will be made partner in their law firm. Deborah guesses her probability is 0.7 and that Matt’s is 0.5 Are these events disjoint? Deborah also guesses that the probability that she and Matt will both be made partners is 0.3 What is the probability that at least one of them will be promoted? P(at least one promoted) = 0.7 + 0.5 – 0.3 = 0.9 What is the probability that neither of them will be promoted?

  20. Independence and the Multiplication Rule • Suppose we toss a coin twice and count the number of heads, so the two events of interest are: • A = first toss is a head • B = second toss is a head • Are A and B disjoint?

  21. P(2 heads) = ? • What is the probability of a head on the first toss? ½ • What is the probability of a head on the second toss? ½ Because the first toss doesn’t change the probability of the second toss we say that these events are independent. • P(2 heads) = (½)(½) = ¼

  22. More Probability Rules • Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent we apply the multiplication rule: P(A and B) = P(A)∙P(B)

  23. Independent Events • You draw 2 cards out of a standard deck of 52 playing cards without replacement. Is the event that you will draw 2 reds independent? • If your doctor takes your blood pressure twice, are the two results independent? • If you take the ACT twice, are the results independent? NOTE: Disjoint events cannot be independent.

  24. Conditional Probability • Conditional Probability • When the probability we assign to an event can change if we know that some other event has occurred. • Amarillo Slim wants an ace, if no cards have been dealt, what is the probability that he will be dealt an ace? 4/52 = 1/13 • If Amarillo already has 1 ace in his 4-card hand, what is the probability that he will be dealt an ace? 3/48 = 1/16

  25. Conditional Probability • The notation for a conditional probability is: P(A|B) read “probability of A given B” • In the last example we would write: P(ace|1 in 4 visible cards) = 1/16 • P(A|B) = P(A and B) P(B)

  26. Conditional Probability • Look at Table 6.1 on page 442 • A = the grade comes from and EPS course • B = the grade is below a B • P(B) = 3656/10,000 = 0.3656 • P(A) = 1600/10,000 = 0.16 • P(B|A) = 800/1,600 = 0.5 • P(A|B) = 800/3,656 = 0.2188

  27. General Multiplcation Rule • Intersection - When all of the events occur. • What if events A and B are not independent? • Two events A and B are independent if P(B|A) = P(B) • General Multiplication Rule for Any Two Events: P(A and B) = P(A ∩ B) = P(A)∙P(B|A)

  28. General Multiplicatio Rule • Slim is still playing cards, now he wants to draw 2 diamonds in a row. He has 4 cards in his hand and there are 7 upturned cards on the table. Of the 11 visible cards there are 4 diamonds. Find Slim’s probability of drawing 2 diamonds. • P(next 2 cards diamonds) = (9/41)∙(8/40) = 0.044

  29. Extended Multiplication Rules • Extending the multiplication rule: • P(A and B and C) = P(A)∙P(B|A)∙P(C|A and B)

  30. Future Athletes • Only 5% of male high school basketball, baseball, and football players go on to play at the college level. Of these, only 1.7% become professional athletes. About 40% of the athletes who reach the pros have a career of more than 3 years. A = {competes in college} B = {competes professionally} C = {pro career longer than 3 years}

  31. A = {competes in college} B = {competes professionally} C = {pro career longer than 3 years} • What is the probability that a high school athlete competes in college and then goes on to have a pro career of more than 3 years? • P(A) = 0.05 • P(B|A) = 0.017 • P(C|A and B) = 0.40 • P(A and B and C) = (0.05)(0.017)(0.40) =

  32. Rules of Probability • Review:

  33. Given two events, E and F, such that P(E) = .340, P(F) = .450, and P(E U F) = .637, then the two events are • Independent and mutually exclusive. • Independent, but not mutually exclusive. • Mutually exclusive, but not independent. • Neither independent nor mutually exclusive. • There is not enough information to answer this question.

  34. If P(A) = .25 and P(B) = .34, what is P(A U B) if A and B are independent? • .085 • .505 • .590 • .675 • There is insufficient information to answer this question.

  35. Given the probabilities P(A) = .3 and P(A U B) = .7, what is the probability P(B) if A and B are mutually exclusive? If A and B are independent? • .4,.3 • .4,4/7 • 4/7,.4 • .7,4/7 • .7,.3

  36. There are five outcomes to an experiment and a student calculates the respective probabilities of the outcomes to be .34, .50, .42, 0, and -.26. The proper conclusion is that • The sum of the individual probabilities is 1 • One of the outcomes will never occur. • One of the outcomes will occur 50 percent of the time. • All of the above are true. • The student made an error.

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