EXAMPLE 1 Construct a probability distribution

Slides:

Advertisements

Similar presentations

1. Frequency Distribution & Relative Frequency Distribution 2. Histogram of Probability Distribution 3. Probability of an Event in Histogram 4. Random.

Advertisements

EXAMPLE 3 Construct a binomial distribution Sports Surveys

6.2 Construct and Interpret Binomial Distributions

Describing Probability

Random Variables.

Working with Random Variables. What is a Random Variable? A random variable is a variable that has a numerical value which arises by chance (ie – from.

Section 7.4 (partially). Section Summary Expected Value Linearity of Expectations Independent Random Variables.

PROBABILITY  A fair six-sided die is rolled. What is the probability that the result is even?

EXAMPLE 1 Construct a probability distribution Let X be a random variable that represents the sum when two six-sided dice are rolled. Make a table and.

Random Variables November 23, Discrete Random Variables A random variable is a variable whose value is a numerical outcome of a random phenomenon.

Probability Mass Function Expectation 郭俊利 2009/03/16

Solve an equation with variables on both sides

EXERCISE R.4 1 R.4*Find the expected value of X in Exercise R.2. [R.2*A random variable X is defined to be the larger of the numbers when two dice are.

PROBABILITY WITH APPLICATION Random outcome  Example of rolling a die Repeating an experiment  Simulation using StatCrunch Software Probability in Real.

EXAMPLE 1 Find probabilities of events You roll a standard six-sided die. Find the probability of (a) rolling a 5 and (b) rolling an even number. SOLUTION.

Binomial PDF and CDF Section Starter Five marbles are on a table. Two of them are going to be painted with a “W” and the rest will be painted.

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 1 of 15 Chapter 12 Probability and Calculus.

Unit 6 – Data Analysis and Probability

THE BINOMIAL DISTRIBUTION. A Binomial distribution arises in situations where there are only two possible outcomes, success or failure. Rolling a die.

Probability Distributions. Essential Question: What is a probability distribution and how is it displayed?

1 Northern Arizona University Department of Mathematics and Statistics 2014 High School Math Day Face Off Students, start your brain cells!

Section 5.1 Expected Value HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights.

III. Probability B. Discrete Probability Distributions

ENGG 2040C: Probability Models and Applications Andrej Bogdanov Spring Random variables part one.

4.1 Probability Distributions. Do you remember? Relative Frequency Histogram.

Chapter 7: Probability Lesson 6: Probability Distributions Mrs. Parziale.

Chapter 5.1 Probability Distributions.  A variable is defined as a characteristic or attribute that can assume different values.  Recall that a variable.

Warm Up a) 41 b) Alternative c) 14/41 = 34%. HW Check.

Lesson Objective Understand what we mean by a Random Variable in maths Understand what is meant by the expectation and variance of a random variable Be.

Mutually Exclusive and Inclusive Events

4.1 Probability Distributions NOTES Coach Bridges.

Warm-up 6.1 (Day 1) Random Variables and Expected Values

EXAMPLE 4 Find probabilities of complements Dice When two six-sided dice are rolled, there are 36 possible outcomes, as shown. Find the probability of.

The Binomial Distribution

4.2 Binomial Distributions

Discrete Distributions. Random Variable - A numerical variable whose value depends on the outcome of a chance experiment.

Modeling Discrete Variables Lecture 22, Part 1 Sections 6.4 Fri, Oct 13, 2006.

Lesson Discrete Random Variables. Objectives Distinguish between discrete and continuous random variables Identify discrete probability distributions.

EXAMPLE 1 Find a normal probability SOLUTION The probability that a randomly selected x -value lies between – 2σ and is the shaded area under the normal.

AP STATISTICS Section 7.1 Random Variables. Objective: To be able to recognize discrete and continuous random variables and calculate probabilities using.

Geometric Distribution. The geometric distribution computes the probability that the first success of a Bernoulli trial comes on the kth trial. For example,

Section 5.2: PROBABILITY AND THE NORMAL DISTRIBUTION.

Probability Distributions and Expected Value

Check it out! : Standard Normal Calculations.

Solutions to the HWPT problems. 1.How many distinguishable arrangements are there of the letters in the word PROBABILITY? Answer: 11!/(2! 2!) 2.A and.

Which list of numbers is ordered from least to greatest? 10 –3, , 1, 10, , 1, 10, 10 2, 10 – , 10 –3, 1, 10, , 10 –3,

EXAMPLE Ordering Real Numbers Order the numbers 0.47, 0.474, 0.47, and 0.23 from least to greatest. SOLUTION STEP 1 Write each decimal out to six.

Probability Distributions Section 7.6. Definitions Random Variable: values are numbers determined by the outcome of an experiment. (rolling 2 dice: rv’s.

Copyright © Cengage Learning. All rights reserved. 8 PROBABILITY DISTRIBUTIONS AND STATISTICS.

Basics of Multivariate Probability

Geometric Distribution

Random Variables and Probability Distribution (2)

Uniform Distributions and Random Variables

Suppose you roll two dice, and let X be sum of the dice. Then X is

Probability Review for Financial Engineers

The Binomial Distribution

Binomial Distributions

Warm Up Imagine you are rolling 2 six-sided dice. 1) What is the probability to roll a sum of 7? 2) What is the probability to roll a sum of 6 or 7? 3)

Lecture 23 Section Mon, Oct 25, 2004

Discrete Distributions

Probability The risk of getting struck by lightning in any year is 1 in 750,000. The chances of surviving a lightning strike are 3 in 4. These risks.

Discrete Distributions

Uniform Distributions and Random Variables

Modeling Discrete Variables

Discrete Distributions.

Objective: Find the mean, variance, and standard deviation of a discrete random variable Standard DA-5 Lesson 5.2.

6.1 Construct and Interpret Binomial Distributions

Modeling Discrete Variables

Chapter 11 Probability.

Presentation transcript:

EXAMPLE 1 Construct a probability distribution Let X be a random variable that represents the sum when two six-sided dice are rolled. Make a table and a histogram showing the probability distribution for X. SOLUTION The possible values of X are the integers from 2 to 12. The table shows how many outcomes of rolling two dice produce each value of X. Divide the number of outcomes for X by 36 to find P(X).

EXAMPLE 1 Construct a probability distribution

EXAMPLE 2 Interpret a probability distribution Use the probability distribution in Example 1 to answer each question. What is the most likely sum when rolling two six-sided dice? What is the probability that the sum of the two dice is at least 10? SOLUTION The most likely sum when rolling two six-sided dice is the value of X for which P(X) is greatest. This probability is greatest for X = 7. So, the most likely sum when rolling the two dice is 7.

EXAMPLE 2 Interpret a probability distribution The probability that the sum of the two dice is at least 10 is: P(X > 10 ) = P(X = 10) + P(X = 11) + P(X = 12) = 3 36 + 2 1 = 6 36 = 1 6 0.167

GUIDED PRACTICE for Examples 1 and 2 A tetrahedral die has four sides numbered 1 through 4. Let X be a random variable that represents the sum when two such dice are rolled. Make a table and a histogram showing the probability distribution for X.

GUIDED PRACTICE for Examples 1 and 2 A tetrahedral die has four sides numbered 1 through 4. Let X be a random variable that represents the sum when two such dice are rolled. What is the most likely sum when rolling the two dice? What is the probability that the sum of the two dice is at most 3? SOLUTION The most likely sum when rolling two four-sided dice is the value of X for which P(X) is greatest. This probability is greatest for X = 5. So, the most likely sum when rolling the two dice is 5.

GUIDED PRACTICE for Examples 1 and 2 P(X ≤ 3 ) = P(X = 2) + P(X =3) = 1 16 + 8 = 3 16 sum of 5; 3 16 ANSWER