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Basics of Probability

Basics of Probability. Trial or Experiment. Experiment - a process that results in a particular outcome or “event”. Simple event (or sample point), E i – an event that can’t be decomposed into multiple individual outcomes.

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Basics of Probability

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  1. Basics of Probability

  2. Trial or Experiment • Experiment - a process that results in a particular outcome or “event”. • Simple event(or sample point),Ei– an event that can’t be decomposed into multiple individual outcomes. • Sample space, S- The set of all possible sample points for the experiment. • Event, Ai - a subset of the sample space.

  3. Likelihood of an Outcome • define the "likelihood" of a particular outcome or “event”where an event is simply a subset of the sample space. • Assuming each sample point is equally likely,

  4. A Simple Experiment • jar contains 3 quarters, 2 dimes, 1 nickels, and 4 pennies, • consider randomly drawing one coin.The sample space: • Let A be the event that a quarter is selected

  5. Drawing a Quarter? • Randomly draw a coin from the jar... • There are 3 quarters among the 10 coins: • Assuming each coin is equally likely to be drawn.

  6. Roll the Dice • Using the elements of the sample space: (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) • Considering the sum of the values rolled,

  7. Roll the Dice • Using the elements of the sample space: (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) • Count the members for this event.

  8. Roll the Dice • Using the elements of the sample space: (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) Likelihood one of the faces shows a “2” ? P( a “2” is rolled ) =

  9. Properties of a Probability • Each outcome Ai in the sample space is assigned a probability value P( Ai ) 1). “between a 0% and a 100% chance of occurring”: 2). Summing over all the sample points in the sample space… “a 100% chance one of the outcomes occurs” and…

  10. Properties of a Probability …and 3). When a set of events A1, A2, A3,… is pairwise mutually exclusive… P( “2” is rolled OR sum is greater than 8 ) =

  11. The Sample-Point Method • Define the sample space: describe and list the simple events, being careful not to include any compound events. • Assign a probability to each sample point, satisfying the “properties of a probability”. • Define the event of interest, A, as a set of sample points. • Compute P(A) by summing the probabilities of sample points in A.

  12. Sticky Spinner 2 1 3 6 • Suppose a game uses a spinner to determine the number of places you may move your playing piece. • Suppose the spinner tends to stop on “3” and “6” twice as often as it stops on the other numbers. • What is the probability of moving a total of 9 spaces on your next 2 spins? 4 5

  13. Multiplication Principle ( called “mn rule” in text )

  14. By the multiplication principle, if | A| = n, then Cross Product and Power Set • By the multiplication principle,if | A| = m and | B| = n, then | A x B| = mn.

  15. Printer2 choices Scanner4 choices “Decision Tree” Total of 24 different systems Computer3 choices

  16. Addition Principle For any two sets A and B, In particular, if A and B are disjoint sets, then

  17. Extended to 3 sets… May generalize further for any n sets.

  18. So for probability… • Leads to an “addition rule for probability”:

  19. Additive Rule of Probability and if events A and B are mutually exclusive events, this simplifies to

  20. Either Way • Note we can do addition first, then convert to a probability ratio: • Or we can construct the probabilities,then do addition:

  21. Compute the Probability • (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) • (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) • (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) • (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) • (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) • (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

  22. sedan mini-van truck totals male .16 .10 .20 .46 female .24 .22 .08 .54 .40 .32 .28 1.00 Compute the Probability • Given the following probability table: • If one of the owners is randomly selected…

  23. Counting Permutations “Permutations of n objects, taken r at a time” The number of ways to choose and arrange any r objects chosen from a set of n available objects, when repetitions are not allowed.

  24. Or equivalently, Gold, Silver, Bronze • Consider the top 3 winners in a race with 8 contestants. How many results are possible?

  25. Calculate it • Calculators have a built-in feature for these computations (labeled as nPr ). • Use the MATH button and PRB submenu. • To compute the value we simply enter:8 nPr 3

  26. Compare the 2 cases Consider a club with 16 members: • Case 1: • If a president, VP, and treasurer are elected, how many outcomes are possible? • (select and arrange 3, order is important) • 16 x 15 x 14 = 3360 pres. VP treas. • Case 2: • If a group of 3 members is chosen, how many groups are possible ? • (a choice of 3 members, order is not important) • Since we don't count the different arrangements, this total should be less.

  27. Adjust the total Case 1: Case 2:Given one group of 3 members, such as Joe, Bob, and Sue, 6 arrangements are possible:( Joe, Bob, Sue), ( Joe, Sue, Bob), ( Bob, Joe, Sue) ( Bob, Sue, Joe), ( Sue, Joe, Bob), ( Sue, Bob, Joe)Each group gets counted 6 times for permutations. Divide by 6 to “remove this redundancy”.

  28. Counting Combinations • “Combinations of n objects, taken r at a time”when repetitions are not allowedOften read as “n, choose r" Sometimes denoted as

  29. All Spades? • For example, in a 5-card hand, P( all 5-cards drawn are spades)

  30. 4 Spades, and a Non-Spade? • For example, in a 5-card hand, P(exactly 4 spades in 5-card hand)?

  31. All Possible Cases? Consider the possible number of spades: • P(all 5 spades) = 0.00049520 • P(exactly 4 spades) = 0.01072929 • P(exactly 3 spades) = 0.08154262 • P(exactly 2 spades) = 0.27427971 • P(exactly 1 spade) = 0.41141957 • P(no spades) = 0.22153361 1.000

  32. Exactly 3 Face Cards? • “3 face cards” implies other 2 cards are not face cards • P( 5-card hand with exactly 3 face cards) = ?

  33. Probable Committee? • If a 3-person committee is selected at random from a group of 6 juniors and 9 seniors, what is the probability that exactly 2 seniors are selected? • Setup the ratio, this type of committee as compared to all possible 3-person committees.

  34. Binomial Coefficients • Recall the Binomial Theorem: For every non-negative integer n… Remember “Pascal’s Triangle”?

  35. Multinomial Coefficients • “Ways to partition n objects into k groups”when repetitions are not allowed Here the groups are of size n1, n2, …, and nk such that n1 + n2 + … + nk = n.

  36. In the expansion of the multinomial Determine the coefficient of the x3y3z2 term. Expanding a Multinomial • Using the multinomial coefficients

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