So, here are some exercises that you may want to solve to get from this chapter as much as you can (best option). Alternatively, you may read the task descriptions and the solutions (and try to understand them).
Some mobile phones and cash dispensers prevent unauthorized access to the resources by using a 4-digit PIN number.
What is the probability that a randomly typed number will be the right one?
Hint. Calculate how many different numbers you can type. If you get stuck, try to reduce the problem to 1- or 2-digit PIN number.
A few years ago during a home party a few people bragged that they can recognize beer blindly, just by taste, since, e.g. “the beer of brand X is great, of brand Y is OK, but of brand Z tastes like piss” (hmm, how could they tell?).
We decided to put that to the test. We bought six different beer brands. One person poured them to cups marked 1-6. The task was to taste the beer and correctly place a label on it.
What is the probability that a person would place correctly 6 labels on 6 different beer at random.
Hint. This task may be seen as ordering of different objects. As always you may reduce the problem to a smaller one. For instance think how many different orderings of 3 beer do we have.
Do you still remember our tennis example from Section 4.7.1, I hope so. Let’s modify it a bit to solidify your understanding of the topic.
Imagine John and Peter played 6 games, but this time the result was 1-5 for Peter. Is the difference statistically significant at the crazy cutoff level for \(\alpha\) equal to 0.15. Calculate the probability (the famous p-values) for one- and two-tailed tests.
In the opening to Section 4.7.5 I told you a story from the old times. The day when I met my friend Paul in a local chess club and lost 6 games in a row while playing with him. So, here is a task for you. If we were both equally good chess players at that time then what is the probability that this happened by chance (to make it simpler do one-tailed test)?
Remember how in Section 4.7.5 we talked about a type II error. We said that if we decide not to reject \(H_{0}\) we risk to commit a type II error or β. It is FN, i.e. false negative, in our judge analogy from Section 4.7.5 (declaring a person that is really guilty to be innocent). In statistics this is when the \(H_{A}\) is true but we fail to say so and stay with our initial hypothesis (\(H_{0}\)).
So here is the task.
Assume that the result of the six tennis games was 1-5 for Peter (like in Section 4.8.3). Write a computer simulation that estimates the probability of type II error that we commit in this case by not rejecting \(H_{0}\) (if the cutoff level for \(\alpha\) is equal to 0.05). To make it easier use one-tailed probabilities.
Hint: assume that \(H_{A}\) is true and that in reality Peter wins with John on average with the ratio 5 to 1 (5 wins - 1 defeat).