Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Prob and Stats, Oct 9 Probability I – Introduction and the Definitions in Probability, Simple Probability Book Sections: N/A Essential Questions: How can I compute the probability of any event? What is probability and what is the mathematical basis of computing it? How do I compute simple probability? Standards: PS.SPCR.1, .1a, .1b Probability What is probability? • A measure of the likelihood that an event will happen. Simple Probability Terms to Know and Understand • Outcome – The result of a random phenomenon • Random –outcomes that occur by chance, and every outcome has the same chance or is equally likely • Event – A collection of outcomes • Probability – The likelihood that an event will happen • Sample Space – A list of all possible outcomes of an event Notation • The probability of an event will be abbreviated as follows: P(event) = Examples In a coin flip there are two possible outcomes, head and tail The probability of the outcome head of the event would be expressed as: P(head) The probability of the outcome tail of the event would be expressed as: P(tail) More Examples The Mathematical Basis of Probability • You compute probability, no matter how it is expressed, by using a ratio. • A ratio is comparing two numbers by division. The most common manifestation of a ratio is a fraction. Ratio • A ratio is a comparison of two numbers by division • There are three ways of writing a ratio: 5 5 to 22, or 5:22, or 22 Examples 1) 15 cats to 50 dogs 2) 27 nurses to 9 doctors 3) 8 completions:12 passes 4) 21 hired out of 105 applicants 5) 2 teams : 22 players 6) 7 first downs to 28 first downs Expressing Probability • Probability can be expressed as: A fraction (in simplest form) A decimal A percent (%) • Probability ranges between 0 and 1 (0% - 100%) Probability of 0 means the event is impossible Probability of 1 means the event is a sure thing The Range of Probabilities likely What are Favorable Outcomes? • Favorable outcomes of some event A are the ways within the sample space that A can happen. Examples Probability in this Course • We will compute simple and every other type of probability you can think of in this class. You will need to know how to: Create and simplify fractions, add fractions, multiply fractions over the course of this unit. • The calculator can do all of these, understand how and know the calculator limitations! The Mathematical Definition of Probability Number of favorable outcomes P(event) = Total number of outcomes In words: The probability of an event is the ratio of favorable outcomes to the number of possible outcomes. That number will always be between 0 and 1. What Are the Components of the Ratio • Favorable outcomes – The number of ways within the sample space that the event (what you want to occur) CAN occur • Total number of outcomes – everything that can happen in the sample space • In these two counts you are counting the options, the outcomes are not the options – it is the number of outcomes The total number of outcomes is also known as all possible outcomes Some Simple Examples Favorable Outcomes (How Many?) If you: Select the number 5 on a roll of a die. = 1 Select an even number on a roll of a die. = 3 Select a red card from a deck of cards. = 26 Select a queen from a deck of cards. = 4 Select the 5 of diamonds from a deck of cards. = 1 Call heads on a coin flip. = 1 Some Simple Examples Total Number of Outcomes If you roll a single fair die, there are 6 possible outcomes, which are 1, 2, 3, 4, 5, 6 those are all the numbers on the standard cube Drawing a single card from a deck. There are 52 possible outcomes, because there are that many cards in the deck. Flipping a coin has two possible outcomes, a head or a tail, there are two sides on every coin. Computing Probabilities The probability of the number 5 on a roll of a die. The probability of an even number on a roll of a die. The probability of a red card from a deck of cards. The probability of a queen from a deck of cards. The probability of the 5 of diamonds from a deck of cards. The probability of heads on a coin flip. Examples P(4) = P(red) = P(even) = P(odd) = P(not 5) = P(9) = P(4 or 5) = P(< 8) = Examples P(4) = P(red) = P(even) = P(odd) = P(not 5) = P(blue) = P(4 or 5) = P(prime) = Examples A bag contains 3 pink, 2 blue, 5 black, 1 clear and 1 yellow marble. Compute the following probabilities based on The probability a single marble from the bag: P(black) P(yellow) P(green) P(clear) P(not blue) P(yellow or blue) Is it Really Random? Or Not When is an occurrence random? Answer: When every outcome is equally likely. What About These? Any problems here? Complementary Events The compliment of an event A is everything happening except A The compliment of A is called not A and is abbreviated with A (called as A bar) or not A Examples The compliment of heads on a coin flip would be not heads which would be ‘tails’ The compliment of a 5 on a dice roll (not 5) - would be 1, 2, 3, 4, or 6 The compliment of a rainy day forecast for today - would be no rain today Complementary Events An event will either happen or it will not. All possible outcomes that are not an event add up to be the compliment of that event. The sum of the probability of an event and the event’s compliment always add up to 1 or P(event) + P(not the event) = 1, where P(not the event) is the probability of the event’s compliment. In Other Words • The probability of a complimentary event is: P(event) = 1 – P(event) or P(A) = 1 – P(A) Example: What is the probability of not getting a 3 when rolling a die? 1 5 1 – P(3) = 1 6 6 Examples Compute each probability P(not even) P(red) P(not yellow) If the P(odin) = ⅜, what is P(not odin)? Class work: CW 10/9/15, Parts 1&2, All Homework: None