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 \(X((x_1, x_2, x_3, x_4, x_5)) = \sum_{n=1}^5 x_n\).