Download presentation

Presentation is loading. Please wait.

Published byDerick Reynolds Modified over 3 years ago

1
10/1/20151 Math 4030-2a Sample Space, Events, and Probabilities of Events

2
10/1/20152 Random Experiment An experiment is called Random experiment if 1.The outcome of the experiment in not known in advance 2.All possible outcomes of the experiment are known.

3
10/1/20153 Sample space and events (Sec. 3.1) Set of all possible outcomes of an experiment is called sample space We will denote a sample space by S finite or infinite. discrete or continuous Any subset of a sample space is called an event.

4
10/1/20154 Operations on events Union,, “or” Intersections,, “and” Complement,, “not” Mutually Exclusive Events Venn diagram

5
Probability of an event (Sec. 3.3) 10/1/20155 Event A P(A) S

6
Axioms of probability (Sec. 3.4) Axiom 1. 0 ≤ P(A) ≤ 1. Axiom 2. P(S) = 1 Axiom 3. If A and B are mutually exclusive events then P(A U B) = P(A) + P(B) 10/1/20156

7
Axioms of probability Generalization of Axiom 3. If A 1, A 2, …, A n are mutually exclusive events in a sample space S then P(A 1 U A 2 U … U A n ) = P(A 1 ) + P(A 2 ) + … + P(A n ) 10/1/20157

8
Addition rule of probability If A and B are any events in S then P(AUB) = P(A) + P(B) – P(A B) Special case: if A and B are mutually exclusive, then P(AUB) = P(A) + P(B). 10/1/20158

9
Probability rule of the complement 10/1/20159 If B is the complement of A, then P(B) = 1 - P(A).

10
Classical probability has assumptions: There are m outcomes in a sample space (as the result of a random experiment); All outcomes are equally likely to occur; An event A (of our interest) consists of s outcomes; Then the definition of the probability for event A is 10/1/201510

11
Relative frequency approach Perform the experiment (trial) m times repeatedly; Record the number of experiments/trials that the desired event is observed, say s; Then the probability of the event A can be approximated by 10/1/201511

12
1210/1/2015 Count without counting: Sample Space Event Pr(Event) = Count!

13
1310/1/2015 Multiplication Rule (P50): k stages; there are n 1 outcomes at the 1 st stage; from each outcome at ith stage, there are n i outcomes at (i+1)st stage; i=1,2,…,k-1. Total number of outcomes at kth stage is

14
1410/1/2015 Permutation Rule (P51): n distinct objects; take r (<= n) to form an ordered sequence; Total number of different sequences is

15
1510/1/2015 Factorial notation: Permutation number when n = r, i.e.

16
1610/1/2015 Combination Rule (P52): n distinctive object; take r (<= n) to form a GROUP (with no required order) Total number of different groups is

17
Count without counting: 10/1/201517 Multiplication:Independency between stages; Permutation: Choose r from n (distinct letters) to make an ordered list (words). Special case of multiplication; Factorial: Special case of permutation; Combination: Choose r from n, with no order.

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google