An Introduction to Probability Theory

Simple Probability Simple probability is defined as:

A probability of an event is certain to occur is 1 and the probability that an event cannot occur is 0. If the probability of an event occurring is p, then the probability of this event not occurring is (1 – p) 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 6 10 14 4 12 20 28 18 30 42 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 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 c) Blue Answer: Answer: b) Yellow d) Not blue Answer: Answer:

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 . Find: c) P(A’) a) P(A) Answer: Answer: b) d) Answer: Answer:

A and B are subsets of the universal set and n(A) = 25, n(B) = 20, and there are 50 elements in the universal set. When one element is selected at random, calculate: a) P(A) c) P(B’) Answer: Answer: b) d) Answer: Answer:

A number k is chosen from {-3, -2, -1, 0, 1, 2, 3, 4} 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? Answer: A number c is chosen from {1, 2, 3, 4, 5, 6}. What is the probability that the expression below intersects the x-axis? Answer:

a) Each letter is in the correct envelope 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: Two dice are rolled and the product of the scores is found. Find the probability that the product is: a) odd b) prime Answers:

Two dice are rolled. Find the probability that: 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: Two dice are rolled. Find the probability that: a) the total score is 10 b) the dice show the same number Answers:

(a) two girls and two boys; 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