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 $A$.

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 $99.9\%$?

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 $7$?