3.2 Conditional Probability and Independence

Transcription

1 3.2 Conditional Probability and Independence Mark R. Woodard Furman U 2010 Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

2 Outline 1 Conditional Probability 2 Independence 3 Examples and Assignment Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

3 Conditional Probability Main Idea The main idea of conditional probability is that knowing some extra information about an experiment might change the probability in a specific way. Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

4 Conditional Probability Main Idea The main idea of conditional probability is that knowing some extra information about an experiment might change the probability in a specific way. Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

5 Conditional Probability Main Idea The main idea of conditional probability is that knowing some extra information about an experiment might change the probability in a specific way. For example, if you pick a card out of a deck at random, the probability that the card is from the diamond suit is 1/4, but if you knew somehow that the card was red, then the probability would jump to 1/2. Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

6 Conditional Probability Main Idea The main idea of conditional probability is that knowing some extra information about an experiment might change the probability in a specific way. For example, if you pick a card out of a deck at random, the probability that the card is from the diamond suit is 1/4, but if you knew somehow that the card was red, then the probability would jump to 1/2. We say that the conditional probability of diamond given red is 1/2. Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

7 Conditional Probability Main Idea The main idea of conditional probability is that knowing some extra information about an experiment might change the probability in a specific way. For example, if you pick a card out of a deck at random, the probability that the card is from the diamond suit is 1/4, but if you knew somehow that the card was red, then the probability would jump to 1/2. We say that the conditional probability of diamond given red is 1/2. The symbolism for this is Pr(A B) which we read as the conditional probability of A given B, or sometimes we simply say the probability of A given B. Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

8 Conditional Probability Main Idea The main idea of conditional probability is that knowing some extra information about an experiment might change the probability in a specific way. For example, if you pick a card out of a deck at random, the probability that the card is from the diamond suit is 1/4, but if you knew somehow that the card was red, then the probability would jump to 1/2. We say that the conditional probability of diamond given red is 1/2. The symbolism for this is Pr(A B) which we read as the conditional probability of A given B, or sometimes we simply say the probability of A given B. In this symbolism, A and B are events, that is, subsets of the sample space. Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

9 Conditional Probability Definition The formula for conditional probability is as follows: Pr(A B) = Pr(A B). Pr(B) Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

10 Conditional Probability Definition The formula for conditional probability is as follows: Pr(A B) = Pr(A B). Pr(B) Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

11 Conditional Probability Definition The formula for conditional probability is as follows: Pr(A B) = Pr(A B). Pr(B) In class, we will dicuss why this is a reasonable formula, but you can take this to be the definition of conditional probability if you like. Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

12 Conditional Probability Definition The formula for conditional probability is as follows: Pr(A B) = Pr(A B). Pr(B) In class, we will dicuss why this is a reasonable formula, but you can take this to be the definition of conditional probability if you like. Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

13 Independence Independent Events The following statements turn out to be equivalent: Whenever any of the above are true, we say that A and B are independent events. Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

14 Independence Independent Events The following statements turn out to be equivalent: Pr(A B) = Pr(A) Pr(B). Whenever any of the above are true, we say that A and B are independent events. Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

15 Independence Independent Events The following statements turn out to be equivalent: Pr(A B) = Pr(A) Pr(B). Pr(A B) = Pr(A). Whenever any of the above are true, we say that A and B are independent events. Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

16 Independence Independent Events The following statements turn out to be equivalent: Pr(A B) = Pr(A) Pr(B). Pr(A B) = Pr(A). Pr(B A) = Pr(B). Whenever any of the above are true, we say that A and B are independent events. Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

17 Independence Independent Events The following statements turn out to be equivalent: Pr(A B) = Pr(A) Pr(B). Pr(A B) = Pr(A). Pr(B A) = Pr(B). Whenever any of the above are true, we say that A and B are independent events. Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

18 Independence Independent Events The following statements turn out to be equivalent: Pr(A B) = Pr(A) Pr(B). Pr(A B) = Pr(A). Pr(B A) = Pr(B). Whenever any of the above are true, we say that A and B are independent events. Warning Note that although the word independent seems something like the word disjoint, they are very different in this context. In particular, if A and B are disjoint, then Pr(A B) = Pr(φ) = 0, rather than as in the definition above. Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

19 Independence Independent Events The following statements turn out to be equivalent: Pr(A B) = Pr(A) Pr(B). Pr(A B) = Pr(A). Pr(B A) = Pr(B). Whenever any of the above are true, we say that A and B are independent events. Warning Note that although the word independent seems something like the word disjoint, they are very different in this context. In particular, if A and B are disjoint, then Pr(A B) = Pr(φ) = 0, rather than as in the definition above. Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

20 Independence Independent Events The following statements turn out to be equivalent: Pr(A B) = Pr(A) Pr(B). Pr(A B) = Pr(A). Pr(B A) = Pr(B). Whenever any of the above are true, we say that A and B are independent events. Warning Note that although the word independent seems something like the word disjoint, they are very different in this context. In particular, if A and B are disjoint, then Pr(A B) = Pr(φ) = 0, rather than as in the definition above. You should think of events as being independent when knowing that one occurs doesn t have any effect on the probability of the other. Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6

21 Examples and Assignment Examples and Assignment Examples: 6, 10, 12, 18, 24. Assignment: Numbers 1, 5, 9, 11, 15, 19, 23, 27, 31. Mark R. Woodard (Furman U) 3.2 Conditional Probability and Independence / 6