Probability 02

Exercise 0.1

A researcher records the daily number of minutes 12 students spent on an online learning platform during the week. The observed data (in minutes) are:

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import math
 
x = [18, 25, 22, 30, 27, 21, 26, 29, 24, 23, 28, 95]
mean = sum(x) / len(x)
median = sum(sorted(x)[5:7]) / 2
s2 = sum((xi-mean)**2 for xi in x) / (len(x) - 1)
s = math.sqrt(s2)

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Exercise 0.2

A performance engineer measures the response time (in milliseconds) of a web service over 20 consecutive requests. The recorded response time are:

Throughout this exercise, use a fixed horizontal axis from to for all histograms and boxplot.

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Exercise 0.3

A city studies its commuters. Each commuter uses exactly one of the following modes of transportation

• Car
• Bus
• Bicycle These three categories are mutually exclusive and cover all commuters. The city also records whether a commuter was delayed (event ). The following information is known:
• of commuters drive a car.
• Among bus commuters, were delayed.
• Overall, of all commuters were delayed.
• Among commuters who were not delayed, used the bus.

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Exercise 0.4

Independent trials resulting in a success with probability and failure with probability are performed. What is the probability that a total of successes occur before a total of failures? (The occurrence of these successes and failures doesn’t have to be consecutive!)

Exercise 0.5

Twins can be either identical or fraternal. Identical, also called monozygotic, twins form when a single fertilized egg splits into two genetically identical parts. Consequently, identical twins always have the same set of genes. Fraternal, also called dizygotic, twins develop when two eggs are fertilized and implant in the uterus. The genetic connection of fraternal twins is no more or less the same as siblings born at separate times. A Los Angeles County, California, scientist wishing to know the current fraction of twin pairs born in the county that are identical twins has assigned a county statistician to study this issue. The statistician initially requested each hospital in the county to record all twin births, indicating whether or not the resulting twins were identical. The hospitals, however, told her that to determine whether newborn twins were identical was not a simple task, as it involved the permission of the twins’ parents to perform complicated and expensive DNA studies that the hospitals could not afford. After some deliberation, the statistician just asked the hospitals for data listing all twin births along with an indication as to whether the twins were of the same sex. When such data indicated that approximately 64 percent of twin births were same-sexed, the statistician declared that approximately 28 percent of all twins were identical. How did she come to this conclusion?

Exercise 0.6

Consider 3 urns. Urn contains 2 white and 4 red balls, urn contains 8 white and 4 red balls, and urn contains 1 white and 3 red balls. If 1 ball is selected from each urn, what is the probability that the ball chosen from urn was white given that exactly 2 white balls were selected?

Exercise 0.7

An urn initially contains 5 white and 7 black balls. Each time a ball is selected, its color is noted and it is replaced in the urn along with 2 other balls of the same color. Compute the probability that

• (a) the first 2 balls selected are black and the next 2 are white;
• (b) out of the first 4 balls selected, exactly 2 are black.

Exercise 0.8

There are 3 coins in a box. One is a two-headed coin, another is a fair coin, and the third is a biased coin that comes up heads 75 percent of the time. When one of the 3 coins is selected at random and flipped, it shows heads. What is the probability that it was the two-headed coin?