Unit 01

A parent receives presents from their three children, Allison, Bobby, and Chelsea. To avoid disputes they open the presents in a random order. Write down a suitable outcome space to represent this situation. Then compute how many events would exist (taking the event space to be the power set).

Prove that if an event space has infinitely many events, the outcome space must have infinitely many outcomes.

Assume a die has numbers 1 through 6 and a probability of $1/6$ of landing on each number. Suppose we roll a red die and a green die (that are independent). What is the probability that the number on the red die is larger than the number on the green die?

a) If we flip a fair coin $5$ times, what is the probability that the number of heads is an even number (note that 0, 2 and 4 are even numbers)?

b) Does your answer depend on the number of flips?

Assume outcome space $\Omega = {a,b}$ (with the usual event space equal to the power set). Prove that the following function is not a valid probability function.

$P(\emptyset) = 0$

$P(a) = .5$

$P(b) = .4$

$P(\Omega) = 1$

Assume an outcome space consisting of all positive integers, $\Omega = {1,2,3,...}$, and a probability function $P$ defined on the power set. Prove that it is not possible for the probability of every integer to be equal. That is it is not possible that $P({a}) = P({b})$ for every pair of integers $a,b \in {1,2,3,...}$.

In a group of students, $25%$ smoke cigarettes, $60%$ drink alcohol, and $15%$ do both. What fraction of students have at least one of these bad habits?

In a group of $320$ people, only $10$ are millionaires but $4$ of the $170$ college graduates are millionaires. How many non-college graduates are not millionaires?

Suppose $A$ and $B$ are disjoint with $P(A)=0.3$ and $P(B)=0.5$. What is $P(A'\cap B')$?

Given two events $A$ and $B$ with $P(A)=0.4$ and $P(B)=0.7$, compute the maximum and minimum possible values for $P(A\cap B)$ and prove that they are the minimum/maximum.

Prove Bonferroni's inequality:
$$P(A\cap B)\geq P(A) + P(B) - 1.$$ Hint: The addition rule and Kolmogorov's axioms.

Suppose that every time a person goes to the pet-shelter, they have a 0.5 chance of taking home a dog and a 0.5 chance of taking home a cat and that all visits are mutually independent. How many times does a person need to go to the pet-shelter to so that the probability that they have both a dog and a cat is $\geq0.95$?

Assume two events, $A$ and $B$ are independent ($A \perp B$). a) prove that $A^C$ and $B^C$ are independent. b) prove that $A$ and $B^C$ are independent.

Let $A$ and $B$ be two independent events with $P(A)=0.4$ and $P(A\cup B)=0.64$. What is $P(B)$?

Roll two dice. Let $A$=“the first die is odd,” $B$=“the second die is odd,” and $C$=“the sum is odd.” Prove that these events are pairwise independent but not mutually independent.

Prove that the empty event $\emptyset$ is independent of $A$ for any event $A$.

A fair coin is tossed three times.
(a) What is the conditional probability of two or more heads given that there was at least one head?
(b) What is the conditional probability given that there was at least on tail?

Suppose there are $n$ events, $A_1,A_2,...,A_n$ and the intersection of all these events is non-empty. Write an inductive proof to show that $$P(A_1\cap A_2\cap\ldots \cap A_n)=P(A_1)P(A_2|A_1)P(A_3|A_1\cap A_2)\ldots P(A_n|A_1\cap A_2\cap\ldots\cap A_{n-1}).$$

In the following diagram of sample space, all angles are right angles, and the probability of each event is proportional to its area. Use the law of total probability to compute the value of $P(B)$:

A box has three coins. One has two heads, one has two tails, and the other is a fair coin with one head and one tail. A coin is chosen at random, is flipped, and comes up heads.
(a) What is the probability that the coin chosen is the two-headed coin?
(b) What is the probability that if the same coin is thrown another time, it will come up heads?

A test for coronaviruses has the following properties: Given that a patient has the coronavirus, the conditional probability of a positive test is .95. Given that a patient does not have the coronavirus, the conditional probability of a negative test is .95 You believe that in the population each patient has a .10 probability of having coronavirus. What is the probability that a person has coronavirus, given they present a positive result?