MATH 350 Practice Exam 01

1. License plates in California (CA) consist of 7 digits: integers $0$ through $9$ inclusive, and letters of the English alphabet (all uppercase). Generally, a license plates has in order one number, three letters (duplicates are allowed), and then three numbers (duplicates are allowed). How many possible license plates are there in CA?
2. A biologist is studying the germination rate of a certain type of plant seed. In controlled laboratory conditions, each seed has an 85% chance of germinating successfully. The biologist plants 120 seeds in a tray.
   1. What is the probability that exactly 100 seeds will germinate?
   2. What is the probability that at most 110 seeds will germinate?
   3. If the biologist conducts another experiment and plants 250 seeds in a large plot, what is the expected number of seeds that will germinate? What is the variance?
3. Consider the NBA player Steph Curry. If Steph Curry attempts $50$ shots in a basketball game and on average will make $45$ of those, how many different ways can Steph Curry make exactly $45$ of $50$ attempted shots during a game? n
4. Assume the components $1, 2, 3, 4, 5$ of the system below are mutually independent, and that the probability each component fails is $p_f$. Calculate the probability that the entire system fails.
5. Assume the components $1, 2, 3, 4, 5$ of the system below are mutually independent, and that the probability each component fail is $p_f$. Calculate the probability that the entire system fails.
6. Suppose your cellphone allows $6$ number PIN codes to unlock it.
   1. Use the product 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?
7. Civil engineers are conducting load tests on a newly constructed bridge. The tests are to determine how many tons the bridge can support. They've performed tests on similar bridges and based on past results, the probability distribution of the maximum load (in tons) the bridge can support is given below. Calculate the expected maximum load the bridge can support based on this distribution.

   Maximum load (tons)	probability
   50	0.1
   55	0.2
   60	0.3
   65	0.2
   70	0.1
   75	0.1

8. Consider the Ligue 1 footballer (French soccer player) Kylian Mbappé. Suppose Mbappé attempts $5$ penalty kicks and has probability $0.9$ of making each penalty kick.
   1. How many different ways can he make exactly $3$ of $5$ attempted penalty kicks?
   2. What is the probability that he makes exactly $3$ of $5$ attempted penalty kicks?
   3. What is the probability that he makes exactly $0$ penalty kicks?
   4. What is the probability that he makes exactly $0$ or exactly $5$ penalty kicks?
9. 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 use math 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.
10. 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?
11. Let $X, Y$ be two random variables representing two standard and fair dice.
    1. What is the name of the distribution used to describe either $X$ or $Y$?
    2. How many possible outcomes are there from rolling the two dice together?
    3. What is the probability of any one pair of values, $x, y$, from rolling the two dice?
    4. How many possible outcomes are there from summing the two dice?
    5. How many ways are there for the random variables to sum to $7$, namely $X + Y = 7$?
    6. What is the probability associated with rolling a $7$, namely $\mathbb{P}[X + Y = 7]$?
    7. Describe the distribution for the random variable $Z = X+ Y$. What is the sample space of $Z$? What is the probability associated with each possible outcome $z$ in the sample space?
    8. What is the expectation of $Z$?
12. Suppose we roll a red die and a green die. What is the probability the number on the red die is larger (>) than the number on the green die?
13. Two students arrive late for a math final exam with the excuse that their car had a flat tire. Suspicious, the professor says “each one of you write down on a piece of paper which tire was flat. What is the probability that both students pick the same tire?
14. Each Google Drive file is identified by a unique ID. Each ID consists of 29 characters, where each character can be an integer from $0$ to $9$ or a letter of the English alphabet (26 of them) in uppercase or lowercase – uppercase letters are considered different than lowercaseletters.
    1. Use the multiplication rule to count the total number of files that Google could store.
    2. If you had a secret Google Drive file, what is the probability that I could guess your file’s ID in one guess?
15. In a particular region, a biologist is studying a rare type of frog. From past studies, it's known that the number of these frogs in a specific pond follows a discrete distribution. Let's say the number of frogs, $X$, has a distribution according to the following table. Suppose that each frog has a certain ecological value associated with it. For instance, every frog helps control insect populations in the pond, and the absence or presence of these frogs can significantly affect the ecosystem balance. Let's say, for every frog present, the ecological benefit or value (in arbitrary units) to the pond is 50 units. What is the expected ecological value provided by the frogs in the pond, based on the given distribution?

    Number of frogs	probability
    0	0.1
    1	0.2
    2	0.3
    3	0.25
    4	0.15

16. Find the expectation of a random variable $X$ which follows the Uniform(a, b) distribution. Hint: $\sum_{i=1}^N i = \frac{N(N+1)}{2}$.