Assignment 4
Indepedence
Suppose you choose a uniformly random number $x$ from the set $\{1,\ldots,n\}$. Consider the following two events:
- $A$: $x$ is divisible by $7$
- $B$: $x$ is divisible by $3$
For which positive integers $n$ are the events $A$ and $B$ independent? A complete answer should give a condition that $n$ must satisfy, a proof that $A$ and $B$ are independent if $n$ satisfies this condition, and a proof that $A$ and $B$ are not independent if $n$ does not satisfy this condition.
Lotto 6/49
A Lotto 6/49 ticket costs \$3. When you buy a ticket you choose a $6$-element subset of the integers $1,2,3,\ldots,49$.
On draw day a machine is loaded with 49 balls with the numbers $1,\ldots,49$ and a random subset of $6$ balls comes out of the machine. If those 6 numbers match the six numbers on your ticket, you win the jackpot.
The value of the jackpot depends on how many people have bought tickets. How large must the jackpot be so that your expected winnings, after accounting for the \$3 you paid for the ticket, is at least $0$?
Friends having lunch
A group of $2n$ friends goes to dinner and they sit evenly spaced around a round table. Each friend tosses a coin and compares the result of their coin toss to that of the person sitting opposite them at the table.
- If the two coin tosses are the same, both friends get to eat.
- If the two coin tosses are different, the friend who tosses the head gets to eat, and the friend who tossed a tail doesn't.
What is the expected number of friends who get food?
Gashapons, again
You have a gashapon machine that contains 1000 horses, 900 of these are red and 100 of these are brown. You spend \$3 and get three random horses from the machine.
What is the expected number of red horses you get?
Fishing a uniform lake.
You go fishing in a lake. There are four types of fish (trout, pike, walley, bass) in the lake. Each time you cast your line, you catch a fish, which is equally likely to be one of these four types. You throw every fish you catch back in the water, so the result of each cast is independent of which types of fish you caught on previous casts. You stop fishing once you've caught at least one fish of each type.
What is the expected number of casts you make before you stop fishing?
Fishing a non-uniform lake
You go fishing in a lake. There are two types of fish (trout and pike) in the lake. Each time you cast your line, you catch a trout with probability $4/5$ and you catch a pike with probability $1/5$. You throw every fish you catch back in the water, so the result of each cast is independent of which types of fish you caught on previous casts. You stop fishing once you've caught at least one trout and at least one pike.
What is the expected number of casts you make before you stop fishing?
Fixed points in random permutations
Let $\pi_1,\ldots,\pi_n$ be a random permutation of the integers $1,\ldots,n$.
What the is the expected number of indices $i\in\{1,\ldots,n\}$ such that $\pi_i=n+1-i$?
Hint: If this seems hard, first try to determine the expected number of indices such that $\pi_i=1$.)
The maximum of two 20-sided dice
You roll two $20$-sided dice, $d_1$ and $d_2$.
What is the expected value of the random variable $Z = \max\{d_1,d_2\}$?
Hint: Remember, from our class on geometric random variables, since $\operatorname{Range}(Z)$ contains only non-negative integers, $E(Z)=\sum_{i=1}^{\infty}\Pr(Z\ge i)$.
Hint: In case it's helpful, you can use this fact: $\sum_{i=1}^{20} i^2 = 2870$.