Arnold is studying the prevalence of three health risk factors, denoted by $A$, $B$, and $C$, within a population of men. For each of the three factors, the probability that a randomly selected man in the population has only this risk factor (and none of the others) is $0.1$. For any two of the three factors, the probability that a randomly selected man has exactly these two risk factors (but not the third) is $0.14$. The probability that a randomly selected man has all three risk factors, given that he has $A$ and $B$ is $\frac{1}{3}$. Find the probability that a man has none of the three risk factors given that he does not have risk factor $A$.

Suppose that one of every $500$ people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For a person who does not have the disease, however, there is a $2\%$ false positive rate--in other words, for such people, $98\%$ of the time the test will turn out negative, but $2\%$ of the time the test will turn out positive and will incorrectly indicate that the person has the disease. Let $p$ be the probability that a person who is chosen at random from this population and gets a positive test result actually has the disease. Find $p$.

A pair of standard $6$-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?

An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $\$20$ or more?

Jean is twice as likely to make a free throw as she is to miss it. What is the probability that she will miss $3$ times in a row?

The probability that Alice can solve a given problem is $1/2$. Beth has $1/3$ chance to solve the same problem. Carol's chance to solve it is $1/4$. If all them work on this problem independently, what is the probability that one and only one of them solves it?

Let $a, b, c, m, n, p, k$ be positive real numbers that satisfy $a+m = b+n = c+p=k$. Show that $an+bp+cm < k^2$.

Given randomly selected $5$ distinct positive integers not exceeding $90$, what is the expected average value of the fourth largest number?

Reimu and Sanae play a game using $4$ fair coins. Initially both sides of each coin are white. Starting
with Reimu, they take turns to color one of the white sides either red or green. After all sides are
colored, the 4 coins are tossed. If there are more red sides showing up, then Reimu wins, and if there
are more green sides showing up, then Sanae wins. However, if there is an equal number of red sides
and green sides, then neither of them wins. Given that both of them play optimally to maximize the
probability of winning, what is the probability that Reimu wins?

Yannick is playing a game with $100$ rounds, starting with $1$ coin. During each round, there is a $n\%$ chance that he gains an extra coin, where $n$ is the number of coins he has at the beginning of the
round. What is the expected number of coins he will have at the end of the game?

The probability of a specific parking slot gets occupied is $\frac{1}{3}$ on any single day. If you find this slot vacant for $9$ consecutive days, what is the probability that it will be vacant on the $10^{th}$ day?

$\textbf{Coin Toss}$

Joe tosses a coin. If he gets heads, he stops, otherwise he tosses again. If the second toss is heads, he stops. Otherwise, he tosses the coin again. The process continues until either he gets heads or $100$ tosses have been done. What is the ratio of heads to tails in all the possible scenarios?

$\textbf{Animal Kingdom}$

In an animal kingdom, there are $n$ carnivores and $m$ herbivores. When two herbivores meet, nothing will happen. When two carnivores meet, both will die. If one herbivore meets one carnivore, the herbivore will die. All such meets can only happen between two animals. All living animals will meet another one sooner or later. If a new animal, either a carnivore or a herbivore, enters this kingdom, what is its probability of survival?

Three ants sit at the three vertices of an equilateral triangle. At the same moment, they all start moving along the edge of the triangle at the same speed but each of them randomly chooses a direction independently. What is the probability that none of the ants collides?

$\textbf{Boys v.s. Girls}$

In a remote town, people generally prefer boys over girls. Therefore, every married couple will continue giving birth to a baby until they have a son. Assuming there is fifty-fifty chance for a couple to give birth to a boy or a girl, what is the ratio of boys to girls in this town over many years?

$\textbf{Mafia}$

You are captured by a mafia. He puts two bullets in adjacent chambers of a standard $6$-chamber revolver. Then he points the gun at your head, and pulls the trigger. You survives. He thinks you may be a lucky man and thus promises to free you if you can survive the second shot. Meanwhile, he also gives you the option to re-spin the revolver before he pulls the trigger again. Should you accept his offer?

$\textbf{Birthday Problem}$

Statistically what is the minimum number of people among which the probability of two people having the same birthday exceeds $50\%$? How about if this probability needs to exceed $99.9\%$?

I can roll a die and collect the amount of money on the die, or if I don't like it, I can roll a second time and I have to pick up the die. What is my expected value?

There is one special coin whose both sides are heads and fifteen regular coins. One coin is chosen at random and flipped, coming up heads.
What is the probability that this coin is the special one?

Randomly draw a card twice with replacement from $1$ to $10$, inclusive. What is the probability that the product of these two cards is a multiple of $7$?