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:
- $H_0$: My machine contains 500 red horses and 500 brown horses.
- $H_1$: 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
- $S=A\cup B$,
- $\Pr(A)=\Pr(B)=2/3$,
- $\Pr(C)=2/5$,
- $\Pr(A\cap C)=\Pr(B\cap C)=1/3$, and
- $\Pr(A\cap B\cap C)=1/6$.
$A$ 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?