4.3 Some Rules of Probability

Transcription

1 4.3 Some Rules of Probability Tom Lewis Fall Term 2009 Tom Lewis () 4.3 Some Rules of Probability Fall Term / 6

2 Outline 1 The addition rule 2 The complement rule 3 The inclusion/exclusion principle Tom Lewis () 4.3 Some Rules of Probability Fall Term / 6

3 The addition rule Theorem (Additivity) Tom Lewis () 4.3 Some Rules of Probability Fall Term / 6

4 The addition rule Theorem (Additivity) If A and B be disjoint events on a common probability space, then P(A B) = P(A) + P(B). Tom Lewis () 4.3 Some Rules of Probability Fall Term / 6

5 The addition rule Theorem (Additivity) If A and B be disjoint events on a common probability space, then P(A B) = P(A) + P(B). In general, if A 1, A 2,..., A k are disjoint events on a common probability space, then P(A 1 A 2 A k ) = P(A 1 ) + P(A 2 ) + + P(A k ). Tom Lewis () 4.3 Some Rules of Probability Fall Term / 6

6 The addition rule A bag of m& m s contains 55 candies with the following distribution of colors: Blue Brown Green Orange Red Yellow An experiment consists of selecting a single candy from bag. Let Bl, Br, G, O, R, and Y be the events of selecting, respectively, a blue, brown, green, orange, red, or yellow candy from the bag. Tom Lewis () 4.3 Some Rules of Probability Fall Term / 6

7 The addition rule A bag of m& m s contains 55 candies with the following distribution of colors: Blue Brown Green Orange Red Yellow An experiment consists of selecting a single candy from bag. Let Bl, Br, G, O, R, and Y be the events of selecting, respectively, a blue, brown, green, orange, red, or yellow candy from the bag. Evaluate P(Bl) and P(Y ). Tom Lewis () 4.3 Some Rules of Probability Fall Term / 6

8 The addition rule A bag of m& m s contains 55 candies with the following distribution of colors: Blue Brown Green Orange Red Yellow An experiment consists of selecting a single candy from bag. Let Bl, Br, G, O, R, and Y be the events of selecting, respectively, a blue, brown, green, orange, red, or yellow candy from the bag. Evaluate P(Bl) and P(Y ). Evaluate the probability that the selected candy is blue or yellow. Tom Lewis () 4.3 Some Rules of Probability Fall Term / 6

9 The addition rule A bag of m& m s contains 55 candies with the following distribution of colors: Blue Brown Green Orange Red Yellow An experiment consists of selecting a single candy from bag. Let Bl, Br, G, O, R, and Y be the events of selecting, respectively, a blue, brown, green, orange, red, or yellow candy from the bag. Evaluate P(Bl) and P(Y ). Evaluate the probability that the selected candy is blue or yellow. Evaluate the probability that the selected candy is brown, green or orange. Tom Lewis () 4.3 Some Rules of Probability Fall Term / 6

10 The complement rule Theorem (The complement rule) For any event E on a probability space, P(E) = 1 P(E c ). Tom Lewis () 4.3 Some Rules of Probability Fall Term / 6

11 The complement rule Theorem (The complement rule) For any event E on a probability space, P(E) = 1 P(E c ). Refer to the m&m data. What is the probability of not selecting an orange candy. Tom Lewis () 4.3 Some Rules of Probability Fall Term / 6

12 The inclusion/exclusion principle Theorem (Inclusion/exclusion) If A and B are any two events on a common sample space, then P(A B) = P(A) + P(B) P(A B) Tom Lewis () 4.3 Some Rules of Probability Fall Term / 6

13 The inclusion/exclusion principle Theorem (Inclusion/exclusion) If A and B are any two events on a common sample space, then P(A B) = P(A) + P(B) P(A B) An experiment consists of selecting a single card from a standard deck of 52 cards. What is the probability of selecting a king or a heart? Tom Lewis () 4.3 Some Rules of Probability Fall Term / 6

14 The inclusion/exclusion principle Theorem (Inclusion/exclusion) If A and B are any two events on a common sample space, then P(A B) = P(A) + P(B) P(A B) An experiment consists of selecting a single card from a standard deck of 52 cards. What is the probability of selecting a king or a heart? Let A and B be events on a common probability space with P(A) =.3, P(B) =.4 and P(A B) =.1. Evaluate the following probabilities: Tom Lewis () 4.3 Some Rules of Probability Fall Term / 6

15 The inclusion/exclusion principle Theorem (Inclusion/exclusion) If A and B are any two events on a common sample space, then P(A B) = P(A) + P(B) P(A B) An experiment consists of selecting a single card from a standard deck of 52 cards. What is the probability of selecting a king or a heart? Let A and B be events on a common probability space with P(A) =.3, P(B) =.4 and P(A B) =.1. Evaluate the following probabilities: Find P(A B) Tom Lewis () 4.3 Some Rules of Probability Fall Term / 6

16 The inclusion/exclusion principle Theorem (Inclusion/exclusion) If A and B are any two events on a common sample space, then P(A B) = P(A) + P(B) P(A B) An experiment consists of selecting a single card from a standard deck of 52 cards. What is the probability of selecting a king or a heart? Let A and B be events on a common probability space with P(A) =.3, P(B) =.4 and P(A B) =.1. Evaluate the following probabilities: Find P(A B) Find P(A B c ) Tom Lewis () 4.3 Some Rules of Probability Fall Term / 6