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An Introduction to Probability Theory Simple Probability Simple probability is defined as: P(event) Number of ways the event can occur Total possible number of outcomes of the trial n( A) P( A) n(U ) A probability of an event is certain to occur is 1 and the probability that an event cannot occur is 0. 0 P( E ) 1 P (U ) 1 If the probability of an event occurring is p, then the probability of this event not occurring is (1 – p) P ( E ') 1 P( E ) P( A) P( A ') 1 If E F , then P( E ) P( F ) E and E’ are Complementary Events Probability (Possibility) Space A number is selected at random from the set {2, 4, 6, 8} and another number is selected from the set {1, 3, 5, 7}. The two numbers are multiplied together. Draw a sample space diagram. 2 2 6 10 14 4 4 12 20 28 6 6 18 30 42 8 8 24 40 56 1 3 5 7 A bag contains 8 disks of which 4 are red, 3 are blue and 1 is yellow. Calculate the probability that when one disk is drawn from the bag it will be: a) Red Answer: c) Blue P(Red) 1 2 Answer: b) Yellow Answer: P(Blue) 3 8 d) Not blue P(Yellow) 1 8 Answer: P(Not Blue) 5 8 You can use Set notation when evaluating probabilities Example: One element is randomly selected from a universal set of 20 elements. Sets A and B are subsets of the universal set and n(A) = 15, n(B) = 10 and n(A B) 7. Find: c) P(A’) a) P(A) Answer: b) P(A) 3 4 Answer: P (A B) Answer: P(A B) d) 7 20 1 P(A') 4 P (A B) Answer: P(A B) 9 10 A and B are subsets of the universal set and n(A) = 25, n(B) = 20, [n(A B) '] 20 and there are 50 elements in the universal set. When one element is selected at random, calculate: a) P(A) Answer: b) c) P(B’) 3 Answer: P(B') 5 1 P(A) 2 P (A B) Answer: P(A B) d) 3 5 P (A B) Answer: P(A B) 3 10 A number k is chosen from {-3, -2, -1, 0, 1, 2, 3, 4}. What is the probability that the expression below can be written as a product of two linear factors, each with integer coefficients? x2 2 x k Answer: 3 8 A number c is chosen from {1, 2, 3, 4, 5, 6}. What is the probability that the expression below intersects the x-axis? y x2 4 x c Answer: 2 3 A secretary has three letters to put into envelopes. Being in a rush, she puts them in at random. Find the probability that: a) Each letter is in the correct envelope b) No letter is in its correct envelope. Answers: 1 6 a) b) 13 Two dice are rolled and the product of the scores is found. Find the probability that the product is: a) odd b) prime Answers: a) b) 16 1 4 Three people, A, B and C play a game which is purely determined by chance. Find the probability that they finish in the order ABC. Answer: 1 6 Two dice are rolled. Find the probability that: a) the total score is 10 b) the dice show the same number Answers: a) b) 16 1 12 There are 30 students in a class, of which 18 are girls and 12 are boys. Four students are selected at random to form a committee. Calculate the probability that the committee contains (a) two girls and two boys; (b) students all of the same gender. Answer: (a) 0.368 (b) 0.130 SPEC06/HL1/15