# Probability Theory. Mohamed I. Riffi. Islamic University of Gaza

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1 Probability Theory Mohamed I. Riffi Islamic University of Gaza

3 Chapter 1 Probability

4 Probability Theorem P(φ) = 0. Theorem If events A and B are such that A B, then P(A) P(B). Theorem For each event A, P(A) 1. Theorem If A and B are any two events, then P(A B) = P(A) + P(B) P(A B). 2

5 Probability A faculty leader was meeting two students in Paris, one arriving by train from Amsterdam and the other arriving by train from Brussels at approximately the same time. Let A and B be the events that the respective trains are on time. If P(A) = 0.93, P(B) = 0.89, and P(A B) = 0.87, find the probability that at least one train is on time. Solution P(A B) = P(A) + P(B) P(A B) = =

6 Probability Theorem If A, B, and C are any three events, then P(A B C) =P(A) + P(B) + P(C) P(A B) P(A C) P(B C) + P(A B C). A survey was taken of a groups viewing habits of sporting events on TV during the last year. Let A = {watched football}, B = {watched basketball}, C = {watched baseball}. The results indicate that if a person is selected at random from the surveyed group, then P(A) = 0.43, P(B) = 0.40, P(C) = 0.32, P(A B) = 0.29, P(A C) = 0.22, P(B C) = 0.20, and P(A B C) =

7 Probability The probability that this person watched at least one of these sports is given as P(A B C) =P(A) + P(B) + P(C) P(A B) P(A C) P(B C) + P(A B C) = =

8 Probability Let a probability set function be defined on a sample space S. Let S = {e 1, e 2,..., e m }, where each e i is a possible outcome of the experiment. The integer m is called the total number of ways in which the random experiment can terminate. If each of these outcomes has the same probability of occurring, we say that the m outcomes are equally likely. That is, P({e i }) = 1, i = 1, 2,..., m. m If the number of outcomes in an event A is h, then the integer h is called the number of ways that are favorable to the event A. P(A) = h m = N(A) N(S). 6

9 Probability Let a card be drawn at random from an ordinary deck of 52 playing cards. Then the sample space S is the set of m = 52 different cards, and it is reasonable to assume that each of these cards has the same probability of selection, 1/52. If A is the set of outcomes that are kings, then P(A) = 4/52 = 1/13 because there are h = 4 kings in the deck. 7

10 Probability HW-1: 1.1-2, 1.1-3, 1.1-4, 1.1-5, 1.1-6, 1.1-7, 1.1-9,

11 Multiplication principle Multiplication Principle: Suppose that an experiment (or procedure) E 1 has n 1 outcomes and, for each of these possible outcomes, an experiment (procedure) E 2 has n 2 possible outcomes. Then the composite experiment (procedure) E 1 E 2 that consists of performing first E 1 and then E 2 has n 1 n 2 possible outcomes. Let E 1 denote the selection of a rat from a cage containing one female F rat and one male (M) rat. Let E 2 denote the administering of either drug A, drug B, or a placebo P to the selected rat. Then the outcome for the composite experiment can be denoted by an ordered pair, such as (F, P). The set of all possible outcomes are: (F, A) (F, B) (F, P) (M, A) (M, B) (M, P) 9

12 Counting techniques Suppose that the experiment E i has n i (i = 1, 2,..., m) possible outcomes after previous experiments have been performed. Then the composite experiment E 1 E 2 E m that consists of performing E 1, then E 2,..., and finally E m has n 1 n 2 n m possible outcomes. Suppose that n positions are to be filled with n different objects. By the multiplication principle, there are possible arrangements. n(n 1) (2)(1) = n! 10

13 Counting techniques Definition Each of the n! arrangements (in a row) of n different objects is called a permutation of the n objects. The number of permutations of the four letters a, b, c, and d is clearly 4! = 24. However, the number of possible four-letter code words using the four letters a, b, c, and d if letters may be repeated is 4 4 = 256, because in this case each selection can be performed in four ways. 11

14 Counting techniques The number of possible ordered arrangements of size r, (r n), taken from n different object is np r = n(n 1)(n 2) (n r + 1). np r = n(n 1) (n r + 1)(n r) (3)(2)(1) (n r) (3)(2)(1) = n! (n r)!. Definition Each of the n P r arrangements is called a permutation of n objects taken r at a time. 12

15 Counting techniques The number of possible four-letter code words, selecting from the 26 letters in the alphabet, in which all four letters are different is 26P 4 = (26)(25)(24)(23) = 26! = 358, ! The number of ways of selecting a president, a vice president, a secretary, and a treasurer in a club consisting of 10 persons is 10P 4 = = 10! 6! =

16 Counting techniques Definition If r objects are selected from a set of n objects, and if the order of selection is noted, then the selected set of r objects is called an ordered sample of size r. Definition Sampling with replacement occurs when an object is selected and then replaced before the next object is selected. By the multiplication principle, the number of possible ordered samples of size r taken from a set of n objects is n r when sampling with replacement. A die is rolled seven times. The number of possible ordered samples is 6 7 = 279,

17 Counting techniques An urn contains 10 balls numbered 0, 1, 2,..., 9. If 4 balls are selected one at a time and with replacement, then the number of possible ordered samples is 10 4 = 10, 000. Note that this is the number of four-digit integers between 0000 and 9999, inclusive. Definition Sampling without replacement occurs when an object is not replaced after it has been selected. By the multiplication principle, the number of possible ordered samples of size r taken from a set of n objects without replacement is n(n 1) (n r + 1) = n! (n r)! = n P r. 15

18 Counting techniques The number of ordered samples of 5 cards that can be drawn without replacement from a standard deck of 52 playing cards is (52)(51)(50)(49)(48) = 52! = 311, 875, ! Definition Each of the n C r unordered subsets is called a combination of n objects taken r at a time, where ( ) n r! nc r = = r r!(n r)!. 16

19 Counting techniques The number of possible 5-card hands (in 5-card poker) drawn from a deck of 52 playing cards is ( ) 52 52C 5 = = 52! = 2, 598, !47! The numbers ( n r) are frequently called binomial coefficients, since they arise in the expansion of a binomial. (a + b) n = n r=0 ( ) n b r a n r. r Note that ( ) ( ) n n! = r r!(n r)! = n! n (n r)!r! =. n r 17

20 Counting techniques suppose an urn contains 3 red balls and 7 black balls. How many ordered ways are there to draw the balls (without replacement) when we only care about the color of the balls? Solution This is a question of finding the positions of the 3 red balls: an unordered choice of 3 out of 10 positions. The black balls will go into the remaining 7 positions. There are ( 10 3 ) such arrangements. 18

21 Counting techniques Definition Each of the n C r permutations of n objects, r of one type and n r of another type, is called a distinguishable permutation. A coin is flipped 10 times and the sequence of heads and tails is observed. The number of possible 10-tuplets that result in four heads and six tails is ( ) 10 4 = 10! 4!6! = 10! 6!4! = ( 10 6 ) =

22 Counting techniques Suppose that in a set of n objects, n 1 are similar, n 2 are similar,..., n s are similar, where n 1 + n n s = n. Then the number of distinguishable permutations of the n objects is ( ) n = n 1, n 2,..., n s n! n 1!n 2! n s!. This is sometimes called a multinomial coefficient. Among nine orchids for a line of orchids along one wall, three are white, four lavender, and two yellow. The number of different color displays is then ( ) 9 = 9! 3, 4, 2 3!4!2! = HW-2: 1.2-1, 1.2-3, 1.2-5, 1.2-7, , , ,

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