Difficulty: easy. Notice we didn't specify which color.
Problem 23
Difficulty: Very clever geometry question.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
Difficulty: pretty easy. Kids will get wrapped up in the P(A) and if-then, but it's an example of a problem that is much easier if you imagine the grid of all possibilities. Definitely a candidate for the SAT.
Problem 27
Difficulty: Medium. It's funny how easily the kids get thrown by an infinite series.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
And then something more complicated, though personally, I'm not a fan of probability questions.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
doesn't mean it's aliens. Just because your feeble brain is stuck on something and can't come up with a reasonable answer, doesn't mean there isn't one.
So here's the moron who's convinced that he sees a cellphone using time-traveler in an outtake from a Charlie Chaplin film.
When faced with two or more conclusions, the simplest one is usually the truth. In this case, the simplest explanation is that this is an alien. No wait, it's a secret plot by time travelers. No wait, she's holding her hand over her ear because it's cold out. No wait, it's DVD distortion. No wait, it's a cellphone. No wait, it's photoshop.
Look, if we can make the entire friggin' Lord of the Ring Trilogy of Monsters, why is "time travel" the first thing this so-called intelligent filmmaker can come up with?
Okay, let's run with a time traveler: Why would someone with the technology to do time travel be wandering around Hollywood with a cellphone that has to be held against the ear and is going to buzz or ringtone at really inopportune times (like that wouldn't be noticed?)? Why not a subdermal subvocalization mic w/ bluetooth connected to a satellite phone to the geosynchronously orbiting time travel pod and the bevy of space-studs dressed in leather thongs? Because that would be ridiculous?
All that technology that I just mentioned is currently available. Even the studs dressed in leather thongs. Cellphones in the form the Irishman witnessed have only been around for 5-6 years. Previously, they were differently shaped and would have required a cellphone tower every three miles. Phones are even now beginning to look different from "this" one - smartphone, bluetooth, etc.
Why would technology from the current, and narrow, 5-6 year window be the one seen? Because this is a self-delusionary fool who has a little imagination and sees what he wants to see.
Or this is a stunt by the Irishman - maybe he wants to get himself a little exposure as an idiot, thinking he'll somehow get more "award-winning" film jobs.
Or this is a stunt by the DVD compiler, trying to sell copies of a boxed set to gullible idiots or people who want to "prove that Irishman wrong," i.e., gullible idiots.
Or this is a misinterpretation of 6 seconds of footage culled from millions of hours of Hollywood video. At this rate, even monkeys can randomly type a line or two of Shakespeare.
To show how little the patrons of the turf know about the theory of odds as practiced at the race track, let readers seek a solution to the following elementary problem: If the odds are 7 to 3 against Apple Pie and 6 to 5 against Bumble Bee, what should be the odds against the famous running horse Cucumber? -Sam Loyd, Cyclopedia of Puzzles, 1914
Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 3, and the host, who knows what's behind the doors, opens another door, say number 1, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors?
First, we have to get beyond the farm kids who say that they wouldn't mind having a goat because a Cadillac wouldn't make it up their driveway anyhow, the kids who think this is a stupid idea for a show and the kids who are sulking in their hoods because this so BORING. Occasionally, I'll change it so the loser doors have a pinata filled with used cat litter - which is often funnier.
"It's fifty-fifty because there's only two doors."
"Lucy, I've got some 'splainin to do."
The statement of the problem is crucial for the students to understand why the problem isn't simple probability, choosing between two doors randomly.
It's because Monty didn't choose randomly that you don't. Monty KNOWS where the prize is. This changes things from "Monty chooses one of two doors randomly" which would make yours a 50-50 choice as well. Instead, it's "Monty's knows" and avoids the real door which messes up the probabilities. Realize that Monty never, in all those years and all those shows, ever opened a door and said "Ooops, there's a car, you lose."
Like for all of you, that never helps with my students either so I try the twenty doors version next. Choose door 1. Monty opens all the doors except number 13. Stay or switch to #13? C O O O O O O O O O O O O X O O O O O O O
Everyone wants to switch. But no one will accept the same reasoning will hold for the 3-door version. "Because we are dealing with only two closed doors. DUH." "It's still 50-50, Curmudgeon."
So I ask the students for a brute force solution. How many different possibilities are there? What are the results? It comes quickly to this:
