department‚ USC‚ Fall 2014 Instructor: Prof. Salman Avestimehr Homework 1 Solutions 1. (Axioms of Probability) Prove the union bound: n P [∪n Ak ] ≤ k=1 P [Aj ]. j=1 The union bound is useful because it does not require that the events Aj be independent or disjoint. Problem 1 Solution We prove this part by induction‚ for k = 2 we have P (A1 ∪ A2 ) = P (A1 ) + P (A2 ) − P (A1 ∩ A2 ) ≤ P (A1 ) + P (A2 ) (1) Now‚ assume that the statement is true for k = n n P (A1 ∪ A2
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probability is the probability that an event will occur given that another has already occurred. If A and B are two events‚ then the conditional probability A given B is written as P ( A | B ) and read as “the probability of A given that B has already occurred.” We are to calculate the probability of the intersection of the events F and G. P(F and G) = P(F) P(G |F) P(F) = 13/40 P(G |F) = 4/13 P(F and G) = P(F) P(G |F) = (13/40)(4/13) = .100 Union of Events P(A or B) = P(A) + P(B) – P(A and
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APStatistics Statistics Assignment 12: Rules of Probability Directions: Complete the assignment on your own paper. Clearly label each answer. (34 points) 1. You roll a pair of standard dice. Create the sample space for a single roll of the dice and use the sample space to compute the following probabilities. (8 points) a. Create a sample space. {1‚ 2‚ 3‚ 4‚ 5‚ 6} b. P (getting a 1 on the first die or getting a 6 on the second die) 1/6+1/6= .333 c. P (getting a 3 on the second die given that
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The water was quiet so quiet that I could hear the ripples in the water. I tried to skip rocks through the water‚ but all I could think about was you. How it was a very sad and lonely day. One day I thought about the trees and the flowers‚ but of course you’d pop up again. I always thought that life would be possible without you. The more I think about it; the more that dream seems to be impossible. I hate this life I live because school nor life could ever be as good as you were. You were the inspiration
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GOALS When you have completed this chapter‚ you will be able to ONEDefine probability. TWO Describe the classical‚ empirical‚ and subjective approaches to probability. THREEUnderstand the terms experiment‚ event‚ outcome‚ permutation‚ and combination. FOURDefine the terms conditional probability and joint probability. FIVE Calculate probabilities applying the rules of addition and multiplication. SIXUse a tree diagram to organize and compute probabilities. SEVEN Calculate a probability using Bayes
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Enterprise Solution Division Random Process In a random process we know that what outcomes or events could happen; but we do not know which particular outcome or event will happen. For example tossing of coin‚ rolling of dice‚ roulette wheel‚ changes in valuation in shares‚ demand of particular product etc. Probability It is the numeric value representing the chance‚ likelihood‚ or possibility a particular event will occur It is measured as the fraction between 0 & 1 (or 0% &100%) Probability
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a particular event will happen if something is done repeatedly‚ (596 Webster’s Dictionary). You cannot determine any events that will happen in the future‚ because there is always a chance that something odd will happen‚ (Linn 39-40). Probability originally started for the purpose and attempt to analyze games of chance. Probability is also used in determining the outcomes of an experiment. Sample space is the collection of all results. Probability is a way to assign every event a value between
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the likelihood of an event happening. Directly or indirectly‚ probability plays a role in all activities. Probability is a measure or estimation of how likely it is that something will happen or that a statement is true. Probabilities are given a value between 0 (0% chance or will not happen) and 1 (100% chance or will happen). The higher the degree of probability‚ the more likely the event is to happen‚ or‚ in a longer series of samples‚ the greater the number of times such event is expected to happen
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coin is tossed (H‚ T). (a) Enumerate the elementary events in the sample space for the die/coin combination. (b) Are the elementary events equally likely? Explain. A) Elementary events are - DIE COIN 1 2 3 4 5 6 HEADS H1 H2 H3 H4 H5 H6 TAILS T1 T2 T3 T4 T5 T6 B) YES‚ EACH EVENT IS EQUALLY LIKELY TO OCCUR. THERE ARE 12 POSSIBLE OUTCOMES AS A RESULT OF ROLLING OE DIE AND FLIPPING ONE COIN‚ THEREFORE THE LIKELYHOOD OF ANY ONE EVENT OCCURING IS 1/12. 5.13 (page 186)‚ 5.13 Given P(A)
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Probability Introduction The probability of a specified event is the chance or likelihood that it will occur. There are several ways of viewing probability. One would be experimental in nature‚ where we repeatedly conduct an experiment. Suppose we flipped a coin over and over and over again and it came up heads about half of the time; we would expect that in the future whenever we flipped the coin it would turn up heads about half of the time. When a weather reporter says “there is a 10% chance
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