## Posts tagged ‘probability’

### Three Outta Four

A cartoon for you on National Egg Day (June 3):

This cartoon was inspired by an article that appeared in the *Salt Lake Tribune*, October 11, 2002:

Bureaucrat’s Math Makes Dizzy Dozenby Paul Rolly and JoAnn Jacobsen-Wells

The menu at the Coffee Garden at 900 East and 900 South in Salt Lake City has included a scrumptious selection of quiche for about 10 years.

The recipe calls for four fresh eggs for each quiche.

A Salt Lake County Health Department inspector paid a visit recently and pointed out that research by the Food and Drug Administration indicates that one in four eggs carries Salmonella bacterium, so restaurants should never use more than three eggs when preparing quiche.

The manager on duty wondered aloud if simply throwing out three eggs from each dozen and using the remaining nine in four‑egg quiches would serve the same purpose.

The inspector wasn’t sure, but she said she would research it.

I promise you, I am not making that up.

### Prime Curio Redux

As I mentioned in yesterday’s review of *Prime Curios*, the book contains a lot of interesting facts about prime numbers. In fact, it contains so many intereting tidbits that I was still reading three hours after posting my review. On page 202, I discovered a rather interesting curio:

968,666,869

The smallest palindromic prime with embedded beast number whose digits contain circles, i.e., using only the digits 0, 6, 8, 9.

What struck me about this curio was the embarassingly small size of the set under consideration. If you consider the four subcategories contained within the description — prime, palindrome, embedded beast number, and using only the digits 0, 6, 8, and 9 — the intersection of those groups is miniscule.

Consider each piece in turn. To limit the discussion, let’s only worry about integers less than one billion, since the prime number described above falls below that threshold.

**Prime**

Rumor has it that there are 50,847,478 prime numbers less than 1,000,000,000. (That value seems reasonable, given that the Prime Number Theorem suggests that there should be about 10^{9}/ln(10^{9}) ≈ 48,254,952.) In other words, about 5.084% of the positive integers up to 1,000,000,000 are prime.

**Palindrome**

In general, there are 9 × 10^{[(n + 1)/2]} palidromes with *n* digits. That means that are there 9 + 9 + 90 + 90 + 900 + 900 + 9,000 + 9,000 + 90,000 + 90,000 = 199,998 palindromes less than 1,000,000,000. In other words, less than 0.020% of those numbers are palindromes.

**Embedded Beast Number**

Analyzing this part was pretty cool. How many numbers less than 1,000,000,000 have an embedded beast number, that is, how many numbers have a string of three consecutive 6’s? It took about an hour of playing to find a general formula. For positive integers with *n* digits, the number of positive integers with an embedded beast number is:

10^{n – 3} + 8 × 10^{n – 4} + (*n* – 4)(9^{2} × 10^{n – 5})

That formula revealed that there are 6,400,000 positive integers with an embedded beast number less than 1,000,000,000, or only about 0.640%.

[**update – 3/16/2011**]

As noted by Joshua Zucker in the comments, this formula is incorrect. It fails to remove numbers that have more than one string of three consecutive 6’s. As Josh noted, there are only 42,291 seven‑digit beast numbers (the formula gives 42,300), and there are only 503,748 eight‑digit beast numbers (the formula gives 504,000). I will try to correct this within the next several days.

**Circle Digits**

If the digits in a number are limited to just 0, 6, 8, and 9, then there are 262,143 positive integers with only circle digits less than 1,000,000,000, or a mere 0.026%.

So, what does all this nonsense get us? It says that the probability of a positive integer less than one billion being a palindromic prime with embedded beast number whose digits contain circles is approximately

*P* = 0.05084 × 0.00020 × 0.00640 × 0.00026 = 0.000 000 000 019 523,

or, in layman’s terms, really frickin’ small.

With the odds at about 1 in 50,000,000,000, it’s no suprise that the first occurrence of this type of number is just shy of a billion at 968,666,869.

### Fair and Square

I recently discovered a great problem:

Three points are randomly chosen along the perimeter of a square. What is the probability that the center of the square will be contained within the triangle formed by these three points?

My colleagues and I spent more time talking about this problem than I care to mention. But when all was said and done, I arrived at a wonderfully elegant solution. As usual, I won’t post the solution now to allow you some time to think about it, but I’ll post it in a few days.

The best part about this problem was the “Aha!” moment it afforded me. The solution eluded me when I forced myself to work on it. But yesterday morning, I was thinking about the problem while walking my dog. No pencil, no paper, no agenda… just time to think. And I kid you not — the solution came to me as I was picking up feces. (I have no idea what that says about me.)

This is my favorite part of mathematics. I can literally spend hours reworking equations, drawing figures, and thinking about a problem, and I’ll make no progress. Then later, when I least expect, when I’m freed from the confines of pencil and paper, the solution gently alights in my mind like a butterfly coming to rest on a marigold.

Oh, how I love that feeling!

Here are some math jokes that involve squares:

What keeps a tree in place?

Square roots.Where is the best location for a multiplication table?

Times Square.

And this one is more of a physics joke than a math joke, but I just love it…

Newton, Leibniz, and Pascal were playing hide-and-seek, and Leibniz was it. Pascal ran into the bushes, but Newton simply drew a box on the ground and stood in the middle of it. When Leibniz finished counting, he turned around and saw Newton just standing there.

“Newton, I’ve found you,” Leibniz said.

“No you haven’t,” argued Newton, “you’ve found Pascal.” Gesturing at the ground, he continued, “One Newton per square meter.”

### The Perfect Pack

I have a quirk.

Okay, truth be known, I have many. But this post is only going to elaborate on a particular mathematical quirk that I have, which involves eating M&M’s. (Most of my other quirks aren’t interesting enough to warrant a blog post. And those that are probably shouldn’t be publicized.)

I have to eat M&M’s in pairs of the same color. I place two in my mouth at a time, and I chew one on each side. I can’t eat them one at a time, and I can’t eat two M&M’s of different colors at the same time. It’s a balance thing. I’ve done this since I was a kid, and whether it’s just a bad habit or a deeply engrained compulsion, I don’t worry about it too much. Sure, it’s a little weird, but on the OCD continuum, it’s not a big deal. I mean, it’s not like I use a ruler to ensure that stamps are placed exactly the same distance from both sides of the envelope. (Though I have considered it.)

So, the math of this. I have been searching for the “perfect pack of M&M’s,” one in which there is an even number of every color. That way, I won’t have one leftover after I eat the others in pairs. I don’t know how many packs of M&M’s I’ve eaten in my life, but I’ve yet to find a perfect pack. Consequently, it would seem that the experimental probability of such an occurrence is 0. But what is the theoretical probability?

Stated formally, here’s the problem:

A standard pack of M&M’s contains pieces of six different colors. What is the probability that there will be an even number of M&M’s of each of the six colors?

I’ll post my solution in a couple days. In the meantime, here are two math jokes about M&M’s:

How many mathematicians does is take to make a batch of chocolate chip cookies?

Two. One to mix the batter, and one to peel the M&M’s.

How do you keep a math graduate student occupied for hours?

Ask him to alphabetize a bag of M&M’s.

### Compound Probability

When I told my friend AJ that I had written a book of math jokes, he asked me a question that I found difficult to answer. He asked, “How many will I laugh at?” I paused for a second. Hearing my hesitation, he asked, “Are the jokes really that bad?”

“Well, no,” I explained. “I’m just not sure how many you’ll get.”

AJ is not a dumb guy. He’s quite intelligent, actually. He can hold his own in a conversation with just about anyone on nearly any topic. But some of the jokes in *Math Jokes 4 Mathy Folks* require some advanced understanding of mathematics. Thinking about his question further, I derived the following formula (though not while I was drinking… I never drink and derive):

P(L) = P(G) × P(F|G)

where…

- P(L) is the probability of laughing;
- P(G) is the probability that you get the joke; and,
- P(F|G) is the probability that you’ll think a joke is funny, if you get it.

The question, of course, is how you determine the values for P(G) and P(F|G). Based on absolutely no data whatsoever, I offer the following:

- P(G) = 0.99, if you have a degree in mathematics;
- P(G) = 0.95, if you completed a high school calculus or statistics course;
- P(G) = 0.68, if you completed the minimum high school requirements in mathematics;
- P(G) = 0.51, if you were reasonably successful in mathematics through middle school;
- P(G) = 0.32, if you were okay until your teachers started using words like
*denominator*and*irrational*; - P(G) = 0.03, if you’re a professional athlete;
- P(G) = 0.02, if you’re a member of my extended family (who hate math, aren’t good at it, and are proud to trumpet both facts to anyone willing to listen);
- P(G) = 0.01, if you’re a journalist or other member of the popular media (and possibly lower, if you write for a tabloid).

Of course, I’m egotistical enough to believe that P(F|G) = 1.

So, how many jokes will AJ laugh at? I don’t know. But with over 400 jokes in *Math Jokes 4 Mathy Folks*, there’s got to be a few that he’ll find funny, right?

Anyway, here’s a joke involving compound probability:

When a respected statistician passed through the security check at an airport, a bomb was discovered in his carry-on luggage. “Come with us,” said the security guards, and they took him to a room for interrogation.

“I can explain,” the statistician said. “You see, the probability of a bomb being on a plane is 1/1000. That’s quite high, if you think about it — so high, in fact, that I wouldn’t have any peace of mind on a flight.”

“And what does that have to do with

youcarrying a bomb on board?” asked a guard.“Well, the probability of

onebomb being on my plane is 1/1000, but the chance of there beingtwobombs on my plane is only 1/1,000,000. So if I bring a bomb, the likelihood of there being another bomb on the plane is very, very low.”

### Baseball Probability

I happen to love sports. I’m a die-hard Pittsburgh Steelers fan, I’ll attend any baseball game I can get tickets to, and don’t even think about calling my house during March Madness or the NBA Finals — I won’t answer the phone.

But if you’re one of those math folks who prefers numbers to games, here’s a primer on recent events, as well as a joke you can tell at the next math department happy hour if the conversation turns to sports.

A perfect game in baseball is one in which a pitcher retires every batter he faces. No players get on base during the entire game — no hits, and no walks. Twenty-seven players come to bat, and all 27 of them make an out.

As you might expect, perfect games are extremely rare. There have been only 20 in major league history.

But recently, they seem to be a little less rare. On Sunday, May 9, pitcher Dallas Braden of the Oakland A’s threw a perfect game against the Tampa Bay Rays. Just 22 days later, Roy Halladay of the Philadelphia Phillies was perfect against the Florida Marlins. And 4 days after that, Armando Galarraga of the Chicago White Sox threw what has been officially ruled a one-hit shut-out… but because of an incorrect call by one of the umpires, most folks think it should count as a perfect game.

Two perfect games in a month is astonishing. Three perfect games in a month is nearly impossible. Don Leypoldt of Hardball Times claims that the odds of throwing three perfect games in a month are approximately 2,000,000 to 1.

But perfect games are not the most rare events in baseball. **Can you name at least two single-game events that are less likely?** (Of course, some events are completely impossible — such as a cow hitting a home run off a curveball thrown by a left-handed pig because, as we all know, *every pig is right-handed*. But by “events that are less likely,” I’m referring to events that have happened more than once, are considered extraordinary, and just aren’t commonplace.) Answers follow the joke below… and I should probably mention that there are way more than two.

The best pitcher on the baseball team failed his math mid-term. His coach, distraught at the possibility of losing his star player, cut a deal with the professor. In the locker room, the coach explained the situation to the pitcher.

“I was able to convince your math professor,” the coach began, “that if you could answer one math question correctly, you wouldn’t have to miss any games. So I’m going to ask you one question, and I need you to focus. If you answer it correctly, you can play in tomorrow’s big game. But if you miss it, you’re academically ineligible until your grades improve.”

“Okay, coach,” the player said. “I’ll do my best.”

“Great,” the coach said. “Here’s your question: What is 2 + 2?”

The player thought for quite some time. Finally, he said hesitantly, “Um, 4?”

“Really?” the coach asked excitedly. “Really? Did you really just say that 2 + 2 is 4?”

Upon hearing this, the other players in the locker room screamed out, “Aw, c’mon, coach… give him another chance!”

A lot of events in baseball are more rare than perfect games:

**Losing a perfect game on the 27th batter.**Armando Galarraga can apparently take solace — he’s the tenth player in MLB history to whom this has happened. But with only 10 occurrences, losing a perfect game on the last batter is more rare than pitching a perfect game.**The unassisted triple play.**There have only been 15 in Major League history. But like perfect games, they’ve been less rare recently — Troy Tulowitzki completed one in 2007, Asdrubal Cabrera had another in 2008, and Eric Brunlett recorded a game-ending unassisted triple play in 2009.**Four or more home runs by the same player in a single game.**Only 15 of these, too, just like unassisted triple plays. No player has ever hit five or more home runs in a game.**Grounding into four double-plays in a single game.**Joe Torre is the only one to hold this distinction.**Stealing six or more bases in a game.**Two players stole 7 bases in a game: George Gore (1881) and Billy Hamilton (1894). Five other players have stolen six, a feat that Eddie Collins of the Philadelphia Athletics accomplished twice (on September 11, 1912, and then again 11 days later on September 22, 1912).**Three or more triples in a game.**George Streif (1885) and Bill Joyce (1897) both had 4 triples in a game, and 12 players have had 3 triples in a game.**Twenty strike-outs in a nine-inning game.**Only three have done this, the most recent Kerry Wood in 1998.

There are a lot of single-game records that are more rare than perfect games. If you’re interested in other crazy baseball statistics and records, check out the Play Index at Baseball-Reference.com.