PA-1 / Lesson 3
For all of the problems below, you need to provide a convincing explanation/motivation for your answer. Note: There are a few problems unrelated to this lesson, but they are related to the previous lesson(s). This is how problem sets will always be organised, as this allows you to practise spotting the key idea or recalling things you have done even out of context.
Problem 1.
More fatalities occurred because of plane crashes in 1910 than now. Does this mean that planes are now more dangerous than before?
Problem 2.
It was reported that the number of milk cows in the United States in 1860 was three times smaller than in 1936. The picture below was used as a cover picture for this report. What do you think about it, why is it strange and deceiving?
Problem 3.
You pick a card from a usual 52-card deck at random. What is the probability that it's a red card? A king? Red king?
Problem 4.
In 1942 Mr. Dewey was elected a Governor. Now, a few years later, he and his team of politicians want to be reelected. In order to advertise themselves, they said the following: "When Dewey was elected Governor in 1942 the minimum teacher's salary in some districts was as low as 900 \$ a year. Today the school teachers in New York enjoy the highest salaries in the world, thanks to his policies. As a result of his policies, the minimum yearly salaries of teachers in New York City now range from 2500 \$ to 5325 \$." What do you think about this statement? Does it prove anything good about Mr. Dewey?
(It was an actual campaign statement (part of it) from a Republican Party in October 1948.)
Problem 5.
Two people roll a fair die once. What is the probability that the first person will get a number strictly larger than the second person? What if the die has the shape of an icosahedron, i.e. it has 20 faces and numbers $1,2,...,20$ written on them? Please specify the probability model you are working in (i.e what are the simple events and their probabilities).
Imagine a person tossing a coin 10 times until they get 10 heads in a row. Sure, it will take a while to get this result, but at some point, they will get it. The moment they get this, they call you up and show the video of them achieving this (obviously, everything before is cut away). You may congratulate them on achieving this, but this video will surely not convince you that any coin is biased or even that their coin is biased. This video will not really affect your conclusions about any of the coins.
Interestingly, a similar mistake was made by most of the articles before the 19th century. I.e. back then, scientists seldom published the results of their experiments. Instead, they described their conclusions and published data that "demonstrated" the truth of those claims. Only later, people started putting more effort into doing proper experiments and statistical analysis. This was the time when statistics basically revolutionised many areas of research, including, for example, agricultural science. "Revolutionised" in a good sense of this word, statistics is not always bad.