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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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4.2 Binomial Distributions

4.2 Binomial Distributions

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