Practice Exam 01

1. Suppose your cellphone allows 6 number PIN codes to unlock it.

   1. Use the multiplication rule to count the total number of available PINS.

   2. If stole your phone, what is the probability that I could guess your PIN in one guess?

2. Consider the sample space \(S = \{-3, -11, 31\}\). Suppose the random variable \(X\) defined on \(S\) has the density function defined below. What is the expectation of \(X\), \(\mathbb{E}[X]\)?

   \(x\)	\(-3\)	\(-11\)	\(31\)
   \(f(x)\)	\(0.2\)	\(0.7\)	\(0.1\)

3. Suppose \(S = \{1, 2, 3\}\), and we try to define a distribution by \(\mathbb{P}[\{1, 2, 3\}] = 1\), \(\mathbb{P}[\{1,2\}] = 0.7\), \(\mathbb{P}[\{1,3\}] = 0.5\), \(\mathbb{P}[\{2, 3\}] = 0.7\), \(\mathbb{P}[\{1\}] = 0.2\), \(\mathbb{P}[\{2\}] = 0.5\), and \(\mathbb{P}[\{3\}] = 0.3\). Is \(\mathbb{P}\) a valid probability function? Why or why not?

4. Consider the Ligue 1 footballer (French soccer player) Kylian Mbappé. If Kylian Mbappé attempts 5 penalty kicks, how many different ways can Kylian Mbappé make exactly 3 of 5 attempted penalty kicks?

5. Consider the random experiment in which a Ligue 1 footballer attempts 5 penalty kicks, where each kick is either a made shot or not. Suppose you are only interested in the total number of made shots.

   1. Describe the sample space, \(S\). You can usemath symbols, sentences, or a combination of both.

   2. Create a reasonable random variable that assigns elements of the sample space \(S\) to non-negative integers, \(X: S \to \mathbb{N}\). Describe what \(X\) does.