$\newcommand{\R}{\mathbb{R}}\DeclareMathOperator{\E}{\mathbf{E}}\DeclareMathOperator{\deg}{deg}\newcommand{\N}{\mathbb{N}}$

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.

Warn­ing: This is not such an easy ques­tion, and fig­ur­ing out the right char­ac­ter­i­za­tion takes time. Don't spend too much time on this ques­tion until you've an­swered the oth­ers, which are con­sid­er­ably eas­ier.

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

Note: The hint in this ques­tion has been up­dated.

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: $\sum_{i=0}^{19} i^2 = 2470$.