When the basketball practice ends, Audrey comes to the room where Pablo and his teammates have been studying for Algebra II. Not only did the footballers go over the long questions that they used, but they also answered a few sample multiple-choice questions in ACT style on probability and statistics.
"Thank you, Pablo. Now we have a better idea of what we need to work on" a grateful Ethan then walks away, with a string of multiple-choice answers and the test's long questions on hand.
Pablo then turns in Finn's direction. "If I may, don't look at the solutions without attempting the problem yourself first!"
"Oh..." a surprised Finn then rushes to the exit, and then kisses Carrie when he gets to her. "Good luck, Pablo, for this... quiz bowl tournament!"
Pablo then blushes. "For once that you have been nice to me about quiz bowl!"
Audrey arrives as Finn and the other football players leave. "Speaking of quiz bowl, some of our teammates are already at my place. Namely, Gerard and Nadine"
The juniors on the quiz bowl team are much more preoccupied with studying for the ACT, and they do so at Myriam's place, Audrey muses while she picks up Carrie alongside Pablo. He knew Carrie in passing, but he never interacted much with her. Maybe in the weight room sometimes but no more than that.
When the trio arrives at Audrey's home, they are arrayed in the dining room, ready to tackle sample probability and statistics questions.
"Carrie, here are some questions from past years' tests on the topic of your test on Monday" Pablo shows her the long questions of the Algebra II test, knowing that Ethan, Finn et al tucked away the answers to the multiple-choice questions.
The first question is about the odds of winning a lottery. However, they don't realize they are solving for a real lottery.
"OK, if we have a six-digit combination that wins you the jackpot, drawn out of forty-nine digits, what's the probability of winning the jackpot?" Pablo asks Carrie about the odds.
"Are the digits drawn with or without return?" a confused Carrie rolls her eyes.
"Did you ever see anyone play a lottery, or a lottery being drawn?" Audrey asks Carrie in hopes of getting her on track to answer.
"No? My parents might have played, but they never showed me the tickets!" a confused Carrie starts wondering what lottery tickets even look like.
Pablo then looks up a ticket for the lottery in the question on his phone. He shows the blank ticket to Carrie:
"This is the ticket of the lottery in the question. To play the lottery, you need to pick six digits, but can you pick the same digit more than once?" he asks her.
"No?" Carrie answers, her tone of voice leaving traces of her insecurity.
"So now you know where to start!" Audrey's eyes are now turned to Carrie's screen.
Carrie starts multiplying what she feels are probabilities of a certain digit being on her ticket. So her worksheet shows 1/49 times 1/48, and so on until all six digits are accounted for. At the end of this stage, she checks her answer against the solution.
"Ouch! My answer is far lower than the solution!" Carrie starts wondering what went wrong in solving the problem.
"There are two questions you should be asking when solving probability problems with combinations. Pablo only hinted at one of the two questions, the other being whether the order of items in a combination is important. Does it matter in which order the digits are drawn?" Audrey's patience nosedives.
To this question, Carrie draws a blank. However, she feels like the second question would be wrong if the first one is since it's about the expected value of playing a lottery.
"Try to imagine how it would be to play a lottery. If you bought a lottery ticket, the receipt would show your combinations in ascending order. The question then becomes: is it important in which order the digits are drawn or not?" Pablo asks a visibly annoyed Carrie.
"I give up..." Carrie laments.
"It's too early for you to give up! I'll even show you a lottery draw! I think it will help you understand!" Audrey then starts a video of a lottery draw.
The other two quiz bowlers in the house start watching the draw video. Then it becomes clear to everyone in the room, even the other two quiz bowlers, that it really doesn't matter how the digits are ordered in the draw. So long as the digits in the winning combination are all drawn in the end.
"When you draw the first ball, there's a six over forty-nine chance that one of the numbers you play will be part of the winning combination, then five over forty-eight..." Audrey explains to her basketball teammate what it means.
"There are intermediate prizes for those who don't win the jackpot, but calculating the probability of winning any of these intermediate prizes is deemed too difficult for you guys over in Algebra II somehow" Gerard looks at the question his teammates are helping Carrie getting through.
The second part of the problem is about the average per-ticket outcome incurred by people playing the lottery, given a prize structure and a table of probabilities. The jackpot's probability is intentionally left blank.
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Here the jackpot is $5 million, and the prize for 2/6 being a free ticket which, for the purpose of the question, is the same as winning $3. There are two sub-questions, and something doesn't sit well with them:
"Why is it that the second sub-question is asking for intermediate information required to answer the first sub-question?" Audrey asks the guests, her mouth wide open.
We all had the experience, one day or another, of being asked about things that are out of logical order on assignments or tests. It seems to be the first time it happens this year in Algebra II this year, though, Pablo seems to be a little... perplexed, while he looks at Carrie trying to solve the problem.
"Think of what is being asked as a weighted average!" Pablo tells Carrie, while Gerard and Nadine get out of the living room. "The weights of the average are the probabilities!"
And then Carrie calculates the weighted average of lottery winnings, to the best of her ability, double-checking her calculations to ensure that she hasn't missed anything. Once the calculations end, Carrie comes to a grim realization:
"It seems that, on average, people win less at the lottery than they spent buying tickets!" Carrie gasps, after subtracting the $3 cost of the ticket to answer the first sub-question. "It makes me not want to play the lottery!"
"And me neither!" Pablo quips, while he prepares a set of ACT questions in the areas of probability and statistics not covered by the long questions, along with the second long question he plans to pass off as a previous year's test item.
I'm not sure if dissuading us from playing the lottery is what the teacher wanted, Pablo is a little perplexed by the implications of the problem's conclusion. However, Audrey's dad interrupts them as they start solving the second long question, which includes a dataset of past VAs' ACT scores.
"I overheard you talk about lotteries. A few more things about lotteries that you might want to consider" he then talks about taxation of gambling income.
He then talks about the tax implications of winning any sum deemed significant by the IRS, with the understanding that winning $10 or less is usually deemed insignificant for tax purposes.
Meanwhile, Carrie receives a text message from Finn. I want you to go to the bathroom during the Algebra II test. It's the only place at school where secret documents can be hidden without anyone noticing.
She starts ruminating. What kind of secret documents could Finn possibly have in his possession that could be of interest to me? His parents don't work for the government, nor in any other field where secrecy is important...
"I think we did a lot for Carrie, I believe that we should take a break for now" Pablo sighs.
"Yeah, I'm hungry... But I think Gerard would be a better tutor for Carrie than we would be!" Audrey can smell her parents cooking Cajun style gumbo, before turning to Gerard. "I believe you might have had some experience tutoring people in math! Maybe you should help Carrie with statistics in our stead!"
"It's always how it is when you're on an academic team: people always ask you for help! But, Audrey, she knows you better than she knows me!" Gerard retorts.
Mathletes are good at math, at least compared to their at-grade peers. However, what makes Audrey think that my knowledge of stats is enough to help Carrie get through a course intended for people two years ahead of me? Does she think that mathletics exposes me to these concepts? A confused Gerard ruminates while he feels Audrey knows, deep down, that he isn't the best mathlete in the world, before turning to Carrie.
"Carrie, I wonder who else you know is a mathlete. Just that, in the future, perhaps asking a different person for help might be helpful" Gerard states, matter-of-factly, how he feels about helping her.
"Other than Myriam and you? I only heard about them by name, but no more than that!" Carrie laments, her face turning red.
While Audrey's parents serve 7 servings of gumbo, they take up all the space available around the dining table. Even then, space is a little tight: there aren't enough chairs for everyone.
"Who's going to sit in the living room?" Audrey's father asks the guests. "Anyone?"
No one seems willing to eat separately from the others, so Audrey's father resigns himself to sitting separately in the living room.
Carrie, on the other hand, is in shock from the topics she hears around the dinner table, which are topics unrelated to basketball. And, for this reason, keeps quiet, feeling out of depth. It's the first time these... study machines calling themselves quiz bowlers invited me for dinner! I don't understand what they're talking about! Finn only talked about sports and exercise with me!
"The southern part of Livonia occupied the northern part of this country" Audrey starts reading a tossup quiz bowl question of her own writing.
The others at the table roll their eyes as Audrey disgorges the clues, which are supposed to get easier as they are being read. First, the Singing Revolution, and then comes the clue that pushes Nadine to answer:
"The Kurzeme Peninsula is..." Audrey keeps reading, interrupted by Nadine.
"Latvia!" Nadine answers.
"Fifteen"
I was only told about Latvia yesterday, and how the star player of Livonia was Latvian; up to this point, it's all I knew about Latvia... Carrie wonders whether quiz bowl is about that sort of obscure knowledge. I think quiz bowl is a stupid game...
"Carrie, now you have a better idea of how quiz bowl questions feel" Audrey explains to her basketball teammate.
"We'll step away from the world of lotteries and into the world of... ACT scores!" Gerard announces to Carrie.
Even though the ACT is still a year away for us, a lot of people in AP Bio are taking it, Pablo ruminates while re-reading the problem statement along with the dataset of a sample of 50 ACT scores from the past five VA classes.
"Box-and-whiskers diagram? What's that?" Carrie gasps, while rolling her eyes at one of the sub-questions.
"First, start with the highest and lowest data points..." Gerard explains to her.
"Carrie, if you dream of going to college, you will see colleges publish the middle fifty percent of admits for both GPA and ACT scores!" Audrey's father explains to them.
Notwithstanding that popular colleges among VAs not in the top 10% often include colleges that have automatic admission plans... Audrey's father refers to McNeese and UL-Lafayette, but doesn't feel like it's a good idea to mention that to her.
"What do the middle fifty percent have to do with a box-and-whiskers diagram?" Carrie asks.
"The middle fifty percent are the edges of the box in a box-and-whiskers diagram!" Gerard answers her, blushing.
"Did basketball-induced fatigue get the better of you? Especially in the past two weeks, as the race to the playoffs intensified?" Nadine asks the small forward.
"I guess..."
However, there's a large number of sub-questions, four of which involve percentiles. What makes it painful to Audrey's parents is that Carrie doesn't seem to even remember how to use a percentile! And has to be explained that aspect of statistics pretty much from the beginning.
And it seems like Audrey's father adds a lot of asides about the stats aspect of college applications. Which makes even Pablo feel like a college's admissions committee is running a lottery.
"Woah, you're making me feel like applying to college is going to be like playing a lottery!" Pablo yelps.
"The colleges that actually feel like lotteries are mostly for top students!" Audrey's father serves a cryptic warning to the quiz bowlers.