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.

$$10 * 10 * 10 * 10 * 10 * 10 = 10^6$$

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

$$1 / 10^6$$

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 $$X$$, $$\mathbb{E}[X]$$?

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

$$\mathbb{E}[X] = \sum_{x \in S} x * f(x) = -3 * f(-3) + -11 * f(-11) + 31 * f(31) = -3 * 0.2 + -11 * 0.7 + 31 * 0.1$$

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?

One example of why this is not a valid probability function is that $$\mathbb{P}[\{2\} \cup \{3\}] = 0.7$$ and should equal $$\mathbb{P}[\{2\}] + \mathbb{P}[\{3\}]$$ but based on what we are given $$\mathbb{P}[\{2\}] + \mathbb{P}[\{3\}] = 0.5 + 0.3 = 0.8$$ not $$0.7$$.

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?

$$C_{3,5} = \frac{5!}{3!(5 - 3)!}$$

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.

Let $$1$$ denote a made shot and $$0$$ a missed shot. Then

$$S = \{(0, 0, 0, 0, 0), (0, 0, 0, 0, 1), (0, 0, 0, 1, 0), \ldots, (1,1,1,1,1) \}$$.

1. 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.

The random variable $$X$$ might assign values by counting the total number of made shots, $$X((0, 0, 0, 0, 0)) = 0$$, $$X((0, 0, 0, 0, 1)) = 1$$, $$X((0, 0, 0, 1, 1)) = 2$$, $$X((1, 1, 1, 1, 1)) = 5$$.

If we wanted to be really math notation heavy, maybe we’d write

$$S = \{(x_1, x_2, x_3, x_4, x_5) : x_n \in \{0, 1\}\}$$. Then $$X((x_1, x_2, x_3, x_4, x_5)) = \sum_{n=1}^5 x_n$$.