# 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.