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Stats for Engineers: Lecture 3

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Conditional probability Suppose there are three cards: A red card that is red on both sides, A white card that is white on both sides, and A mixed card that is red on one side and white on the other. All the cards are placed into a hat and one is pulled at random and placed on a table. If the side facing up is red, what is the probability that the other side is also red? 1.1/6 2.1/3 3.1/2 4.2/3 5.5/6

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Conditional probability Suppose there are three cards: A red card that is red on both sides, A white card that is white on both sides, and A mixed card that is red on one side and white on the other. All the cards are placed into a hat and one is pulled at random and placed on a table. If the side facing up is red, what is the probability that the other side is also red? Red card White card Mixed card Top Red Top White Top Red Let R=red card, TR = top red. Probability tree

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Conditional probability Suppose there are three cards: A red card that is red on both sides, A white card that is white on both sides, and A mixed card that is red on one side and white on the other. All the cards are placed into a hat and one is pulled at random and placed on a table. If the side facing up is red, what is the probability that the other side is also red? The probability we want is P(R|TR) since having the red card is the only way for the other side also to be red. Let R=red card, W = white card, M = mixed card. Let TR = top is a red face. For a random draw P(R)=P(W)=P(M)=1/3. Total probability rule: Intuition: 2/3 of the three red faces are on the red card.

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Summary From Last Time Permutations - ways of ordering k items: k! Ways of choosing k things from n, irrespective of ordering: Random Variables: Discreet and Continuous Mean Means add: Bayes Theorem e.g. from Total Probability Rule:

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Mean of a product of independent random variables Note: in general this is not true if the variables are not independent Example: If I throw two dice, what is the mean value of the product of the throws?

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So the variance can also be written Note that

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Example: what is the mean and standard deviation of the result of a dice throw?

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Sums of variances For two independent (or just uncorrelated) random variables X and Y the variance of X+Y is given by the sums of the separate variances.

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In general, for both discrete and continuous independent (or uncorrelated) random variables Answer: In reality be careful - assumption of independence unlikely to be accurate

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Binomial distribution A process with two possible outcomes, "success" and "failure" (or yes/no, etc.) is called a Bernoulli trial. Discrete Random Variables e.g. coin tossing: Heads or Tails quality control: Satisfactory or Unsatisfactory An experiment consists of n independent Bernoulli trials and p = probability of success for each trial. Let X = total number of successes in the n trials. for k = 0, 1, 2,..., n. This is called the Binomial distribution with parameters n and p, or B(n, p) for short. X ~ B(n, p) stands for "X has the Binomial distribution with parameters n and p." Polling: Agree or disagree

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Situations where a Binomial might occur 1) Quality control: select n items at random; X = number found to be satisfactory. 2) Survey of n people about products A and B; X = number preferring A. 3) Telecommunications: n messages; X = number with an invalid address. 4) Number of items with some property above a threshold; e.g. X = number with height > A

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"X = k" means k successes (each with probability p) and n-k failures (each with probability 1-p). Justification

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Example: If I toss a coin 100 times, what is the probability of getting exactly 50 tails? Answer: Let X = number tails in 100 tosses

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Example: A component has a 20% chance of being a dud. If five are selected from a large batch, what is the probability that more than one is a dud? Answer: Let X = number of duds in selection of 5

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Simple Mathematical Facts for Lecture 1. Conditional Probabilities Given an event has occurred, the conditional probability that another event occurs.

Simple Mathematical Facts for Lecture 1. Conditional Probabilities Given an event has occurred, the conditional probability that another event occurs.

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