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Probability Definition: Probability: the chance an event will happen. Probability = # of ways a certain event can occur # of possible events Probability must be a value between 0 and 1. The probability of the set of all possible outcomes of a trial must be 1 The probability of an event occurring is 1 minus the probability that it does not occur. Complement of A (Ac ) A Rule for Complementary Events: P (A) + P(Ac)= 1 Example: If the probability that a person lives in an industrialized country of the world is 1/5, find the probability that a person does not live in an industrialized country. Answer: P (not living in an industrialized country) = 1- P (living in an industrialized country)= 1 – 1/5= 4/5 Simple Probability Example 1: What is the probability of rolling a 6 on a six sided die? Know: •A six sided die is labeled 1,2,3,4,5,6 •A 6 occurs only once in rolling a die Probability = # of ways a certain outcome can occur # of possible outcomes Probability = 1 6 Example 2 In a statistics class, 32 students of which 20 are females are selected to participate in a study of eye color. It is discovered that 7 of the 32 students have blue eyes. It is also noted that 5 out of the 20 females have blue eyes. Males Blue Eyes No Blue Eyes Total Females Total Answer: Males Females Total Blue Eyes 2 5 7 No Blue Eyes 10 15 25 Total 12 20 32 1. P(Blue eyes) = 7/32 = 0.219 2. P(Females) = 20/32 = 0.625 3. P(Males) = 12/32 = 0.375 4. P(No Blue eyes) = 25/32 = 0.781 Example 4 : In a sample of 50 people, 21 had type “O” blood, 22 had type “A”, 5 had type “B” blood and 2 had type “AB” blood. Set up a frequency distribution and find the following probabilities: 1. A person has type “O” blood. 2. A person has type “A” or type “B” blood. 3. A Person had neither “A” nor type “O” blood. 4. A person does not have type “AB” blood. Example 4 : In a sample of 50 people, 21 had type “O” blood, 22 had type “A”, 5 had type “B” blood and 2 had type “AB” blood. Set up a frequency distribution and find the following probabilities: Group Frequency Type “O” 21 Type “ A” 22 Type “B” 5 Type “AB” 2 Total 50 Answer: Group Frequency Type “O” 21 Type “ A” 22 Type “B” 5 Type “AB” 2 Total 50 1. A person has type “O” blood. P (X = “O”) = 21/50 = 0.42 2. A person has type “A” or type “B” blood. P (X = “A” or “B”) = 26/50 = 0.52 3. A Person had neither “A” nor type “O” blood. P ( X = “B” or “ AB”) = 7/50 = 0.14 4. A person does not have type “AB” blood. P ( X does not have type “AB”) = 48/50 = 0.96 Conditional Probability Formula for Conditional Probability: The probability that the second event B occurs given the first event A has occurred can be found by dividing the probability that both occurred by the probability that the first event has occurred. P( B | A) A P( A and B) P( A) B A and B Example 5 (Conditional Probability) Males Females Total Blue Eyes 2 5 7 No Blue Eyes 10 15 25 Total 12 20 32 What is the probability that a student selected at random is a female given that the student has blue eyes? P(Female | blue eyes) = 5/7 = 0.714 What is the probability that a student selected at random has blue eyes given that the student is male? P(Blue eyes| Male) = 2/12 = 0.167 Compound Probabilities Events that occur in combination • P(blue eyes and female) or in general: P(A and B) Events that occur as alternatives • P(blue eyes or female) or in general: P(A or B) Multiplication (‘AND’) Law Equation #1: If A and B are independent, then; P (A and B) = P(A) x P(B) Equation #2: If A and B are not independent i.e dependent, then; P (A and B) = P (A | B) x P (B) or P (B | A) x P (A) Test of Independent Events: • Two events A and B are independent if the fact that A occurs does not affect the probability of B occurring • Two events A and B are independent events if P(A | B) = P (A) or P(B | A) = P (B) Note :If two events are not independent, they are dependent. Example : In a statistics class, 32 students of which 20 are females are selected to participate in a study of eye color. It is discovered that 7 of the 32 students have blue eyes. It is also noted that 5 out of the 20 females have blue eyes. Males Females Total Blue Eyes 2 5 7 No Blue Eyes 10 15 25 Total 12 20 32 Example : What is the probability of being a female and having blue eyes? Step 1: Is having blue eyes dependent on gender? P(Blue eyes | Female) = P (Blue eye ) 5/20 ≠ 7/32 Thus having blue eyes is dependent on gender. Step 2: Use Equation #2: P (Female and Blue eyes) = P (Blue eyes | Female) x P (Female) = 5/20 x 20/32 = 5/32 = 0.156 Example : A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die. Answer: P (head and 4) = P (head). P (4)= 1/2 * 1/6 = 1/12 = 0.083 Addition (‘OR’) Law Equation #1: If A and B are mutually exclusive: P (A or B) = P(A) + P(B) P(A) (+) P(B) A B A B Equation #2: If A and B are not mutually exclusive: P(A or B) = P(A) + P(B) – {P(A and B)} P(A) P(B) Note: • Two events are mutually exclusive if they cannot occur at the same time (i.e. P(A and B) = 0) A and B Example : In a statistics class, 32 students of which 20 are females are selected to participate in a study of eye color. It is discovered that 7 of the 32 students have blue eyes. It is also noted that 5 out of the 20 females have blue eyes. Males Females Total Blue Eyes 2 5 7 No Blue Eyes 10 15 25 Total 12 20 32 Example : What is the probability a student selected at random will be a female or has blue eyes? Answer: Step 1: Is having blue eyes and gender mutually exclusive? Since a given individual can be a female and have blue eyes, thus they are not mutually exclusive. Step 2: Equation #2: P(Blue eyes or Female) = P(Blue Eyes) + P(Female) – [P(Blue eyes and Female)] = 7/32 + 20/32 – 5/32 = 22/32 = 0.688 Example : A day of the week is selected at random. Find the probability that it is a weekend day. Answer: P (Saturday or Sunday) = P ( Saturday) + P (Sunday) = 1/7 + 1/7 = 2/7 = 0.286