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As­sign­ment 3

The 5-107 Lot­tery

In the 5-107 Lot­tery you choose a set of $6$ dis­tinct in­te­gers $\{x_1,\ldots,x_5,y\}$ from the set $\{1,2,3,\ldots,107\}$. $x_1,\ldots,x_5$ are called your main num­bers and $y$ is your bonus num­ber. On Fri­day night, the lot­tery ma­chine draws a uni­formly ran­dom $5$-num­ber sub­set $\{z_1,\ldots,z_5\}$ from $\{1,\ldots,107\}$. You buy one lot­tery ticket with your favourite $6$ num­bers.

Big Jack­pot

You win a Big Jack­pot if $\{x_1,\ldots,x_5\}=\{z_1,\ldots,z_5\}$. What is the prob­a­bil­ity that you win the Big Jack­pot?

Lit­tle Jack­pot

You win a Lit­tle Jack­pot if $|\{x_1,\ldots,x_5\}\cap\{z_1,\ldots,z_5\}|=4$. What is the prob­a­bil­ity that you win a Lit­tle Jack­pot?

Bonus Jack­pot

If you win a Lit­tle Jack­pot and $y\in\{z_1,\ldots,z_5\}$, then you win a Bonus Jack­pot. What is the prob­a­bil­ity that you win a Bonus Jack­pot?

Ques­tion 2

I have a card game with $100$ cards. For each $i\in\{1,\ldots,100\}$ there is a card with the num­ber $i$ printed on it. I take 5 cards from the deck and get: $\{56, 55, 46, 1, 33\}$. You take a uni­formly ran­dom $5$-el­e­ment sub­set from the re­main­ing cards.

High­est card wins

What is the prob­a­bil­ity that my high­est card (56) is higher than your high­est card?

High­est ver­sus low­est

What is the prob­a­bil­ity that my high­est card (56) is lower than your low­est card?

Sec­ond-high­est ver­sus sec­ond-high­est

What is the prob­a­bil­ity that my sec­ond-high­est card ($55$) is higher than your sec­ond high­est card?

Na­tional-Pub­lic redux

NPR-1

You play a game where you roll an $8$-sided die $8$ times and you win if you roll $8$ at least once. What is the prob­a­bil­ity that you win?

NPR-2

You play a game where you roll an $8$-sided die $16$ times and you win if you roll $8$ at least twice. What is the prob­a­bil­ity that you win?

A Drink­ing Game

Michiel and Pat are on the bal­cony with a cooler that has 20 bot­tles of beer in it: ten bot­tles of lager and ten bot­tles of IPA. They play a game where Pat takes a ran­dom beer from the cooler, drinks it, and throws the bot­tle off the bal­cony, then Michiel does the same. They do this for two rounds, until the neigh­bours call the po­lice.

First IPA Wins (Pat)

In the game of First IPA wins, the per­son who drinks the first bot­tle of IPA is the win­ner (this game can end in a draw if no one drinks an IPA). What is the prob­a­bil­ity that Pat wins this game?

First IPA wins (Michiel)

What is the prob­a­bil­ity that Michiel wins the game of First IPA wins?

Sec­ond IPA Wins (Pat)

In the game of Sec­ond IPA wins, the per­son who drinks the sec­ond bot­tle of IPA is the win­ner. What is the prob­a­bil­ity that Pat wins this game?

Sec­ond IPA Wins (Michiel)

What is the prob­a­bil­ity that Michiel wins the game of Sec­ond IPA Wins?

Are Most Horses Red?

I im­ported a state of the art gashapon ma­chine from Japan that con­tains $1000$ cap­sules, each with a toy horse in it. When I put a dol­lar into the ma­chine it gives me a ran­dom cap­sule that I can open and check the colour of the horse in­side it. The seller claims that half cap­sules con­tain a red horse and half the cap­sules con­tain a brown horse, but I've read re­views from peo­ple com­plain­ing that their ma­chines con­tain 900 red horses and only 100 brown horses.

This leads to two hy­pothe­ses:

1. $H_0$: My ma­chine con­tains 500 red horses and 500 brown horses.
2. $H_1$: My ma­chine con­tains 900 red horses and 100 brown horses.

For any ex­per­i­ment I do, the hy­poth­e­sis $H_0$ de­fines a prob­a­bil­ity func­tion $\Pr_0$ and the hy­poth­e­sis $H_1$ de­fines a dif­fer­ent prob­a­bil­ity func­tion $\Pr_1$.

I only have $\$3$ to fig­ure out which of these hy­pothe­ses is more likely.

A dumb test

Sup­pose I buy three cap­sules from the ma­chine. Let $A$ be the event "the three cap­sules I bought con­tain more red horses than brown horses". Com­pute $\Pr_0(A)$ and $\Pr_1(A)$.

A bet­ter test

De­scribe and an­a­lyze an ex­per­i­ment I can run that costs only $\$3$ and has the fol­low­ing prop­er­ties:

• The re­sult of the ex­per­i­ment is a guess about whether $H_0$ or $H_1$ is true.
• If $H_0$ is true, the ex­per­i­ment guesses $H_0$ with prob­a­bil­ity greater than $1/2$.
• If $H_1$ is true, the ex­per­i­ment guesses $H_1$ with prob­a­bil­ity greater than $1/2$.

Three events

This ques­tion seems to con­tain a cut-and-paste error. Since the as­sign­ment dead­line is so close, you will re­ceive full marks for any an­swer (in­clud­ing no an­swer). Sorry for the trou­ble. -PM

Let $A,B,C\subseteq S$ be three events in a prob­a­bil­ity space $(S,\Pr)$ where

1. $S=A\cup B$,
2. $\Pr(A)=\Pr(B)=2/3$,
3. $\Pr(C)=2/5$,
4. $\Pr(A\cap C)=\Pr(B\cap C)=1/3$, and
5. $\Pr(A\cap B\cap C)=1/6$.

$A$ ver­sus $C$ and $B$ ver­sus $C$

Are the events $A$ and $C$ in­de­pen­dent? Are the events $B$ and $C$ in­de­pen­dent?

$(A\cup B)$ ver­sus $C$

Are the events $(A\cup B)$ and $C$ in­de­pen­dent?