# Worksheet 03

# Probability

Consider the sampe space \(S = \{1, 2, 3, 4\}\). Come up with (create) two different distributions on this sample space. Remember, your distributions must satisfy the Axioms of Probability. You can choose to describe your distributions however you want: assign probabilities to specific sets/subsets of \(S\), draw a or some pictures, or other.

Suppose \(S = \{1, 2, 3, 4, 5, 6, 7, 8\}\) where \(\mathbb{P}[\{s\}] = 1/8\) for \(1 \leq s \leq 8\).

What is \(\mathbb{P}[\{1, 2\}]\)?

What is \(\mathbb{P}[\{1, 2, 3\}]\)?

How many events \(A\) are there such that \(\mathbb{P}[A] = 1/2\)?

# Uniform Distribution

- Suppose you roll 2 fair six-sided dice. What is the probability that the sum of the two dice is equal to 9?

# Permutations

- Imagine a club consisting of 10 members where they need to select a president, a vice president, a secretary, and a treasurer. How many different ways can they select members for these positions, if no person can hold more than one position?

# Combinations

Suppose you roll 5 fair six-sided dice. How many ways can exactly two of the five dice show a 3? What is the probability that exactly two of the five dice show a 3?

How many binary sequences (a sequence of 0s and 1s) of length 10 with exactly four 1s can be formed?

# Random Variables / Density Functions

Consider the sample space \(S = \{dog, cat\}\).

Create a random variable that assigns elements of the sample space to the set \(\{0, 1\}\).

Create a distribution on your random variable from above – assign a probability to each element of the sample space.

Draw a corresponding density function for your distribution.

# Expectation (discrete distributions)

Consider the random variable \(X\) which has a discrete uniform distribution on \(S = \{1, 2, 3, 4, 5, 6\}\), \(X \sim \text{Uniform}(1, 6)\). Recall that the density function of \(X\) is

\[f(x | 1, 6) = 1/6\]

What is the expectation of \(X\)?

Imagine a lottery with only two outcomes, you win or lose. If a ticket costs $1 and a winning lottery ticket wins $1,000,000, what must the probability of winning be in order to break even on average (in expectation)? What happens on average (in expectation) if the probability of winning is anything less than this? What happens on average (in expectation) if the probability of winning is anything more than this?

Consider the two grade distributons from our Syllabus, Grade Distribution X and Grade Distribution Y.

What percentages on each component would a student need such that they earn 100% for their overall grade?

Give an example of a student and their percentages for each component such that they earn exactly 90% for their overall grade?