hw02.pdf - hw02 CS 237 Summer 2019 Homework 02 This homework is due Tuesday 5/28 at 11:59pm with a grace period of 6 hours HW 02 Requirements For the

# hw02.pdf - hw02 CS 237 Summer 2019 Homework 02 This...

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5/28/2019 hw02 localhost:8889/nbconvert/html/Desktop/CS237/hw02.ipynb?download=false 1/11 CS 237 Summer 2019, Homework 02 This homework is due Tuesday 5/28 at 11:59pm with a grace period of 6 hours. HW 02 Requirements For the following problems, analyze means to specify 1. The sample space S, 2. The probability function P, 3. The events specified (i.e.,list the members of each event), and 4. The probability of each of the events. In some cases, additional information may be required. You may abbreviate or schematize as necessary, as long as the answer is perfectly clear. Sometimes it is useful to first write down a "pre-sample space" which helps to think about the actual sample space. This is often useful when the literal outcome of the random experiment is non-numeric but the sample space is numeric. You are not required to submit this pre-sample space in your solution, but I encourage you to take this step whenever possible. Example: Toss a fair coin (i.e., probably of heads is 0.5) and report the number of heads that appear. Let A = "one head appears." Analyze. Solution: The pre-sample space is { T, H }. Then: S = { 0, 1 } P = { 0.5, 0.5 } A = { 1 } P(A) = 0.5 Problem 1 Suppose that a study is being done on all families with 1, 2, or 3 children (all having different ages, i.e., no twins), and let the outcomes be the genders (G = girl and B = boy) of the children in each family in ascending order of their ages (e.g., BG means an older girl and a younger boy). Assume all possible configurations of genders and numbers of children is equally likely (i.e., this will be an equiprobable probability space). Let events A = "families where the oldest child is a boy" and B = "families with exactly two girls and any number of boys." Analyze.

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5/28/2019 hw02 localhost:8889/nbconvert/html/Desktop/CS237/hw02.ipynb?download=false 2/11 Solution: S' = {G, B} P = {0.5, 0.5} S = {G, B, GG, GB, BG, BB, BBB, GGG, BBG, BGB, GBB, GGB, GBG, BGG} => |S| = 14 A = {B, GB, BB, GBB, GGB, BGB, BBB} B = {GG, GGB, GBG, BGG} P(A) = 7/14 = 0.5 P(B) = 4/14 = 0.285 Problem 2 Suppose you ﬂip 3 unfair coins, where the probability of heads is 0.6 and the probability of tails is 0.4, and you count the number of heads showing. Let A = "the number of heads showing is odd" and B = "the number of heads showing is greater than 1." Give the pre-sample space (the three outcomes for each coin, e.g., HTH). Analyze. Solution: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. P(H) = 0.6, P(T) = 0.4 A = {HTT, THT, TTH, HHH} B = {HHT, HTH, THH, HHH} Problem 3 Suppose that each time Wayne charges an item to his credit card, he rounds the amount to the nearest dollar in his records (assume that for x dollars, the amount x.50 is rounded to x + 1 dollars). The round-off error is defined as (recorded - actual); the units are dollars, so if Wayne charges 4.25, he records it as 4, and the round-off error is -0.25, but if he charges 4.75, the value recorded is 5 and the round-off error is 0.25. Assume this is random, so that each time Wayne charges to his card, he performs a random experiment whose outcome is the round-off error. Assume the outcomes are equiprobable.
• Summer '19
• Probability theory, Snyder, GBG

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