Dr. Hanan Mohamed Aly 1 Basic Probability Concepts The concept of probability is frequently encountered in everyday communication. For example, a physician may say that a patient has a chance of surviving a certain operation. Another physician may say that she is 95 percent certain that a patient has a particular disease. Most people express probabilities in terms of percentages. But, it is more convenient to express probabilities as fractions. Thus, we may measure the probability of the occurrence of some event by a number between 0 and 1. The more likely the event, the closer the number is to one. An event that can’t occur has a probability of zero, and an event that is certain to occur has a probability of one.

Dr. Hanan Mohamed Aly2 Objective Probability Classical Probability: For example, in the rolling of the die, each of the six sides is equally likely to be observed. So, the probability that a 4 will be observed is equal to 1/6. Equally Likely Outcomes are the outcomes that have the same chance of occurring. The set of all possible outcomes (The universal set) S contains N mutually exclusive and equally likely outcomes. The empty set φ contains no element.

Dr. Hanan Mohamed Aly3 The event, E is a set of outcomes in S which has a certain characteristic. The probability of the occurrence of E equal to m/N. P(E) = m/N, where m is the number of outcomes which satisfy the event E.

Dr. Hanan Mohamed Aly4 Relative Frequency Probability: If some process is repeated a large number of times, n, and if some resulting event E occurs m times, the relative frequency of occurrence of E, m/n will be approximately equal to the probability of E. P(E) = m/N. Subjective Probability Probability measures the confidence that a particular individual has in the truth of a particular proposition. For example, the probability that a cure for cancer will be discovered within the next 10 years.

Dr. Hanan Mohamed Aly5 Properties of Probability Given some process (or experiment) with n mutually exclusive events E 1, E 2, …, E n, then 1- P (E i ) ≥ 0, i = 1, 2, … n 2- P (E 1 ) + P (E 2 ) + … + P (E n ) = 1 Relations Between Events: 1) Union: A  B means A or B. Example: Let S = {1,2,3,4,5,6,7,8,9,10}, A be choosing an odd number > 2, then A = {3,5,7,9}, P(A) = 0.4 and B be choosing a number divisible by 3, then B = {3,6,9}, P(B) = 0.3. A  B = {3,5,6,7,9} and P(A  B) = 0.5. A B A

Dr. Hanan Mohamed Aly6 2) Intersection: A  B means A and B. Example: In the above example, A = {3,5,7,9}, and B = {3,6,9}, then A  B = {3,9} and P(A  B) = 0.2. A B A ) Complement: A ` means the complement of A, where A  A` = S and A  A` =φ. A  A` = S and A  A` = φ.Example: In the above example, B = {3,6,9}, P(B) = 0.3, then B ` = {1,2,4,5,7,8,10} and P(B ` ) = A B B

Dr. Hanan Mohamed Aly7 Rules of Probability 1- A and B are called disjoint if A  B = , and then P(A  B) = 0 and P(A  B) = P(A) + P(B). For example, if A is choosing an odd number < 11, A = {1,3,5,7,9} and B is choosing an even number < 11, B = {2,4,6,8,10}. Then P(A  B) = 0 and P(A  B) = P(A) + P(B) = If A and B are not disjoint, then P(A  B) = P(A) + P(B) - P(A  B) For example, if A is choosing a number divisible by 5 A = {5,10} and B is choosing an even number < 11, B = {2,4,6,8,10}. Then P(A  B) = 0.1 and P(A  B) = P(A) + P(B) - P(A  B) = A B B A

Dr. Hanan Mohamed Aly8 3- P(A) + P(A ` ) = 1. For example, if A is choosing a number < 5, A = {1,2,3,4}, P(A) = 0.4, but A ` = {5,6,7,8,9,10}, P(A’) = 0.6 Then P(A) + P(A ` ) = P(A) = P(A  B) + P(A  B ` ) For example, if A is choosing a number divisible by 5 A = {5,10}, P(A) = 0.2 and B is choosing an even number < 11, B = {2,4,6,8,10}. Then P(A  B) = 0.1 and P(A  B ` ) = 0.1. Then P(A) = P(A  B) + P(A  B ` ) A B A

Dr. Hanan Mohamed Aly9 5- P(A `  B ` ) = 1 – P(A  B). For example, if A is choosing a number divisible by 5 A = {5,10}, P(A) = 0.2 and B is choosing an even number < 11, B = {2,4,6,8,10}. Then P(A  B) = 0.6 and P(A `  B ` ) = 0.4. Then P(A `  B ` ) = 1 – P(A  B). Two- way Table of Probabilities: BA Total B`B`B`B`BP(A) P(A  B ` ) P(A  B) A P(A`) P(A`  B ` ) P(A`  B) A`A`A`A` P(S) = 1 P(B`) P(B)Total

Dr. Hanan Mohamed Aly10 Calculating The Probability of an Event Example: Here is the data of a sample of adults in a certain city: SumFemale (B ` ) Male(B) 23815Diabetic(A) Normal (A ` ) Sum P(A) = 23 / 125 P(A`) = 1- P(A) = 102 / 125 P(B) = 55 / 125 P(B`) = 70 / 125 P(A  B) = 15 / 125 P(A  B`) = 8 / 125 P(A`  B) = 40 / 125 P(A`  B`) = 62 / 125 P(A) = P(A  B) + P(A  B`) = 23 / 125 P(A  B) = P(A) + P(B) - P(A  B) = 23 / / 125 – 15 / 125 = = 63/ AB

Dr. Hanan Mohamed Aly11 Example: For a sample of 80 recently born children, the following table is obtained: SumGirl(G)Boy(B)Sex Weight in Kg. 734 < 2.5 Kg (L) < 3 (N) (O) Sum P(L) = 7/80 P(N) = 57/80 P(O) = 16/80 P(B) = 49/80 P(G) = P(B`) = 1 - P(B) = 31/80 P(L  B) = 4/80 P(N  B) = 35/80 P(O  B) = 10/80 P(L  G) = 3/80 P(N  G) = 22/80 P(O  G) = 6/80 P(L  B) = 7/ /80 - 4/80 = = 52/80 P(N  B) = 57/ / /80 = = 71/80 P(O  B) = 16/ / /80 = = 55/80 P(L  N) = 7/ /80 = 64/80 P(B  G) = 49/ /80 = 1