4.2 Probability Models

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1 4.2 Probability Models Ulrich Hoensch Tuesday, February 19, 2013

2 Sample Spaces Examples 1. When tossing a coin, the sample space is S = {H, T }, where H = heads, T = tails. 2. When randomly selecting a single digit from the Random Digits Table (Table B), the sample space is S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. 3. When randomly selecting a single person in a town and determining her or his blood type, the sample space is S = {0, A, B, AB}.

3 Events Examples 1. The event of selecting a single odd digit from the Random Digits Table is A = {1, 3, 5, 7, 9}. 2. The event of getting heads and tails when tossing a coin twice is A = {HT, TH}. Note that the sample space is S = {HH, HT, TH, TT }.

4 Probability Rules 1-4

5 Equally Likely Outcomes Examples 1. The probabilities of heads or tails for tossing a fair coin once are P(H) = P(T ) = 1/2. 2. The probability is getting any one of the six numbers when a fair die is rolled once is 1/6.

6 Independence and the Multiplication Rule

7 Examples 1. If a coin is tossed twice, the events A = {H on first toss} and B = {H on second toss} are independent, so P(HH) = P(H)P(H) = (1/2)(1/2) = 1/4. 2. If two cards are drawn at random and without replacement from a standard deck of 52 cards, the events A = {first card is black} and B = {second card is black} are not independent: P(black, black) = = P(black) P(black) The probability that a child in a non-smoking middle class family dies of sudden infant death syndrome (SIDS) is 1 in The probability that in such a family two children die of SIDS is not (1/8500)(1/8500) = 1/72, 250, 000, because multiple occurrences of SIDS are not independent.

8 Example: HIV Testing The probability of a false positive (that is, the probability that a person without the disease tests positive) when using a rapid HIV test is If a hospital tests 200 healthy people using the rapid HIV test, what is the probability that at least one person tests positive? The probability that none of the 200 people test positive is P(no positives) = ( )( ) (1 0.04) }{{} 200 times = (0.996) 200 = %. So, the probability that at least one person tests positive is P(at least one positive) = = = 55.14%.

9 4.3 Random Variables Ulrich Hoensch Thursday, October 4, 2012

10 Random Variables The are two types of random variables: discrete random variables; and continuous random variables.

11 Discrete Random Variables

12 Example: Grade Distribution The grade distribution for the random variable X : the grade of a randomly selected student in an English class is given here. Value of X Probability where A = 4, B = 3, C = 3, D = 1, F = 0. The probability that a randomly selected student scores a B or better is then P(X 3) = P(X = 3) + P(X = 4) = = 0.71.

13 Example: Tossing a Coin Four Times Suppose a fair coin is tossed four times. The sample space is S = {HHHH, HHHT, HHTT,..., TTTT } and the probability of each outcome is (1/2)(1/2)(1/2)(1/2) = 1/16. Consider the random variable X : the number of heads when tossing the coin four times. Then X can be any of the values 0, 1, 2, 3, 4, and we would like to complete the following table. Value of X Probability?????

14 Example: Tossing a Coin Four Times We note that X = 0 corresponds to the event {TTTT }, X = 1 corresponds to {HTTT, THTT, TTHT, TTTH}, etc.

15 Example: Tossing a Coin Four Times So, P(X = 0) = 1/16, P(X = 1) = 4(1/16), etc. and the following table results. Value of X Probability 1/16 4/16 6/16 4/16 1/16 This can be represented graphically using a probability histogram.

16 Example: Uniform Distribution A probability experiment consists of spinning the following arrow.

17 Example: Uniform Distribution Then the sample space is S = {all real numbers x so that 0 x < 1}. The resulting distribution for X : the number the arrow points to, is a uniform distribution. Suppose we want to find P(0.3 X 0.7). This is the area between 0.3 and 0.7 and below the density curve of X. So, P(0.3 X 0.7) = 0.4 and P(X 0.5 or X > 0.8) = 0.7.

18 Continuous Random Variables

19 Example: Grade Distribution Suppose the scores on a statistics exam that is given to a large number of students are normally distributed with mean µ = 72 and standard deviation σ = 15. Let X be the score of a randomly selected student. Find the probability that the student scores 80 points or higher. P(X 80) corresponds to the following area. Using a calculator we can compute P(X 80) = normalcdf(80, , 72, 15) 29.7%. 0

The probability of an event is viewed as a numerical measure of the chance that the event will occur.

Chapter 5 This chapter introduces probability to quantify randomness. Section 5.1: How Can Probability Quantify Randomness? The probability of an event is viewed as a numerical measure of the chance that