Posts tagged ‘probability’
With one week left in the NFL season, Dan Graziano had this to say about the Indianpolis Colts’ chances of making the playoffs:
Indianapolis can still win a third straight AFC South title. Really, it can. All it needs is to win and then have the Texans, Bengals, Chargers, Jets, Saints, Chiefs, Patriots and Browns all lose. The league will throw in the partridge in a pear tree.
If the Colts win and the Houston Texans lose, both would be 8-8, and the first four tiebreakers for deciding which team makes the playoffs – record against one another, record against divisional opponents, record against common opponents, and record within the conference – would not be enough to decide who makes the cut.
It then comes down to strength of victory and strength of schedule. And for things to play out in the Colts’ favor, a lot of things have to go their way.
Fox Sports referred to this as long shot, comparing it to a recent win by a horse who was 200-to-1:
Sure, going 9 for 9 here looks dim, but long shots come in every once in a while.
Are the Colts’ odds as good as that horse’s? Seems not.
Using a simplistic model, assume that each of the nine necessary outcomes are equally likely. That alone would put the Colts’ odds at 511-to-1. (Since 29 = 512.)
But it’s not that simple. The following chart from 538.com gives the probability of each team winning their game this weekend:
The good news is that the Colts have an 81% chance of winning their game against Tennessee. The bad news is that it seems unlikely that any of the Texans, Bengals, Chiefs, or Patriots will lose, let alone all four of them. So putting all those numbers together, the Colts’ chances of making the playoffs are:
0.81 × 0.20 × 0.22 × 0.84 × 0.54 × 0.69 × 0.15 × 0.18 × 0.77 = 0.0002 = 0.02%
or, more precisely, about 4,311-to-1. That’s more than a long shot; that’s an extended-to-an-unfathomable-distance shot.
The Colts are in the unenviable position of Lloyd Christmas in Dumb and Dumber, “So, you’re telling me there’s a chance…”
Is dreidel fair?
The rules of dreidel are straightforward. At the beginning of each round, players put one coin into the pot. (For young kids, the “coins” are actually chocolate pieces in the shape of a coin and wrapped in gold foil. This is known as geld, and as far as I’m concerned, chocolate is a currency to kids.) Players then take turns spinning the dreidel, and a reward is earned based on which of the four Hebrew letters appears on top when the dreidel stops spinning:
- Nun: nothing.
- Hey: half the pot.
- Gimel: all of the pot.
- Shin: put one in.
Play continues clockwise, with each person spinning the dreidel until Gimel occurs and all coins are removed from the pot. At that point, everyone antes another coin, and a new round starts with the next player.
Officially, a player is out of the game when she or he has no coins left to contribute to the pot, and the game ends when one person has all the coins. But practically speaking, the game often ends much earlier, because players get bored and quit or, in the case of very young kids, the game lasts beyond bedtime and the children are pulled away by their parents.
No matter how the game ends, though, it’s not fair.
The following table is courtesy of Paul J. Nahin (Will You Be Alive 10 Years from Now?, Princeton University Press, 2014, p. 81). It shows the amount, over the long run, that each player will win during a dreidel game.
In other words, the first player has a significant advantage over the others. In a game of five players who start with 10 coins each, the first player will finish the game with 19 coins, on average, whereas the fifth player will finish with just 4 coins. That’s if the game ends early. If played until one person gets all the coins, then the first player is five times more likely to win than the fifth player.
This disparity in odds is likely the reason that an unofficial rule of dreidel is that the youngest player goes first, the second-youngest player goes second, and so on.
The word dreidel is Yiddish and means “to turn around.” Because the dreidel is, after all, a top.
“Not My Job” is a segment on the NPR game show Wait, Wait… Don’t Tell Me! During the segment, host Peter Sagal asks a celebrity three questions, on a topic about which they likely have no clue. For instance,
- Cindy Crawford was asked questions about scale models, not supermodels;
- Rob Lowe was asked questions about bratwurst, not the Brat Pack;
- Stephen King was asked questions about the Teletubbies; and,
- Leonard Nimoy was asked questions about the other Dr. Spock (you know, the celebrity pediatrician).
My favorite of these segments, however, was with singer-songwriter Will Oldham, better known by his stage name Bonnie “Prince” Billy. Sagal explained, “You sing mostly sad or mournful songs, interspersed with the occasional tragic one. And we were thinking, who’s the singer least like you? […] So, we’re going to ask you three questions about Ms. Doris Day, the sweet-faced, sweet-voiced singer most famous in the 50s and 60s.”
As Sagal congratulated him, Oldham pretended that Doris Day was in the room with him. “Hey, Doris, you were right!” he said. “All the questions were about you!”
Now, that’s funny!
(As an aside, let me share with you a slide that I sometimes show during presentations about classroom technology:
Yep, that’s Doris Day in the 1958 movie Teacher’s Pet. Hopefully that image looks a little odd to you in 2015. Honestly, if you’re teaching math to adolescents and still using chalk and a wooden pointer, I’ll kindly ask you to consider a different career. There are other options for you. For instance, you could become a dentist and specialize in square root canals. But, I digress.)
So, back to the point.
The celebrity quiz contains three multiple-choice questions, each with three choices. If the guest answers at least two questions correctly, he or she wins. Which got me to thinking…
What is the probability that a celebrity guest will get at least two questions correct if she guesses randomly?
Brute force is definitely an option here. Write down all possible answer choice combinations, randomly decide which configuration will be correct, and then figure out how many of the possible combinations would yield a winner. Not pretty, but it works.
Speaking of combinations, here’s a joke that just has to be shared:
I don’t know if problems like the following are famous, but there sure are a lot of them online — Cut the Knot, Stack Exchange, and Braingle, for example — and they’re typical for a high school classroom or middle school math competition:
There are 14 red, 6 orange, 10 yellow, 8 green, 4 blue, 12 indigo, and 2 violet socks in my sock drawer. How many socks must I randomly remove from the drawer to guarantee that I have two socks of the same color?
You may or may not know the answer, but the problem itself leads to a follow-up question:
Why the hell do I own socks in every color of the rainbow?
That’s just weird. But if you can get past that, here is a related problem:
If I randomly remove two socks from the drawer, what is the probability that they form a matching pair?
As it turns out, I’m something of a sock aficionado. (Yeah, it’s weird, but surely not surprising. I mean, I write a math jokes blog. You didn’t think I was normal, did you?) Although the context above is fictitious, I am indeed the owner of three pairs of identical socks that look like those shown in the picture. And yes, those are my feet and ankles. My mathematical sexy runs all the way down to my toes.
Here’s a close-up of one of them, in case you can’t see it in the larger picture:
That’s a letter R, because these socks are specially designed for each foot. The other sock has a letter L. (Duh.)
This leads to another mathematical question, more real-world than those above:
I just finished washing these three pairs of socks. While folding them, I selected two socks at random and rolled them together. What’s the probability that there’s one R and one L?
The answer, of course, is zero.
Yes, I know that theoretically the answer should be 3/5. But theory doesn’t match practice in this case. When I do my laundry, I sometimes forget to pay attention to the R and the L, and my sock drawer invariably results in one pair of two R’s, one pair of two L’s, and one correctly matched pair. And then when I wake up at 5:30 a.m. and put on my socks in the dark (so as not to rouse my wife from slumber), my feet feel all weird. The one with the wrong sock starts tingling, so I have to remove the socks and choose another pair entirely.
Similarly, here’s another real-world problem, based on my sock experience:
If there are 10 socks in a load of laundry that I place in the washer and then transfer to the dryer, how many socks will remain when the load is finished drying?
Nine. Yes, I know it’s a cliche. Everyone makes jokes about losing socks. It’s so overdone that the National Comedian’s Guild has declared a moratorium against them. But, I’m not joking. I can’t remember the last time I did a load of laundry and wasn’t missing a sock. I now have a drawer filled with unmatched socks, each like Tiger Woods longing for the return of its Lindsey Vonn.
Sadly, this post is going public just a little too late. Lost Socks Memorial Day was May 9, so we just missed that one. Likewise, we missed No Socks Day on May 8. But there are other holidays in the coming months when you can celebrate the amazing undergarments that protect our feet from our shoes:
- July (exact date TBD): Red Socks Day (commemorating Sir Peter Blake)
- October 4: Odd Socks Day (Australia)
- January (every Friday): Snow Sock Day
And though not an official holiday, there are unlimited Crazy Sock Days happening at elementary, middle, and high schools near you.
99% of socks are single, and you don’t see them crying about it.
How do engineers make a bold fashion statement?
They wear their dark grey socks instead of the light grey ones.
Somewhere, all of my socks, Tupperware lids, and ball point pens are hanging out together, just laughing at me.
Because I know you won’t be able to sleep tonight…
- I need to remove 8 socks from my sock drawer to guarantee a color match.
- I don’t actually own socks in every color of the rainbow. Just most colors.
- The probability of selecting two socks from my drawer and getting a matching pair is 23/140.
My favorite question is, “Why?” (And my favorite answer is, “Because.”) But not far behind is the question, “What if?”
What if a baseball player swings a bat with the proper speed, but starts swinging 0.01 seconds too late? What if I could earn 6.3% on a real estate investment instead of 1.4% in a Roth IRA, but had to pay capital gains taxes? What if I tried to walk through a revolving door with a pair of skis on my shoulder?
“What if…?” questions don’t always have to be mathy, ya know.
The beauty of Excel is that you can repeatedly ask “What if…?” questions and then explore to your heart’s content.
Overheard in math class:
“It’s not that I don’t want to do all those math problems,” Julia said to her teacher. “I’m just saying, if we put them into a spreadsheet and let Excel do its thing, we can have an extra 20 minutes for recess.”
Sure, one of the powers of Excel is reducing the tedium associated with calculations, but a much greater power is its ability to allow for deep exploration of math topics in a short period of time.
Art Bardige and Peter Mili agree. That’s why they’re giving away spreadsheets that allow students to explore mathematics.
Their What If Labs allow students to investigate questions like:
- What if you used Excel to design a house?
- Is the world population growing at a faster or slower rate than 50 years ago?
- Instead of wood, nails, and string, what if you used a graph and coordinates to create string art?
The spreadsheets are useful, fun, educational and — dare I say — beautiful. Not to mention, free.
Art believes that teaching math with Excel has two benefits. First, it fosters business skills by having students learn the basics of the most ubiquitous business application on the planet. Second, it empowers students by giving them complete control to explore on their own.
I concur with Art’s philosophy.
Excel is one of my best friends. I use it to test conjectures, especially for probability problems about which I don’t have any intuition — or, more often, when my intuition is wrong!
One of my favorite problems, which was discussed in the post Fair and Square in 2011, is the following:
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?
Would I have solved that problem without Excel? Maybe. Probably. But without Excel, it would have taken longer, and I might not have had the same deep understanding of the underlying structure.
Kudos to Art and Peter for providing a free resource that will let other students benefit from that same type of insight.
Shoes and math have a lot in common.
A shoe salesman consults a mathematician on what size shoes to keep in stock. The mathematician tells him, “There is a simple equation for that,” and shows him the Gaussian normal distribution.
The shoe salesman stares at the equation for a while, then asks, “What’s that symbol?”
“That’s the Greek letter π.”
“What is π?”
“The ratio between the circumference and the diameter of a circle.”
The shoe salesperson thinks for a minute. “What the hell does a circle have to do with shoes?”
As it turns out, there are at least 43 different ways to arrange the laces on your shoes. My favorite is the hexagram method:
But there are some fun things to do with your shoelaces other than lacing up your kicks. Here’s one.
Take the shoelaces out of your shoes. Fold the shoelaces in half and hold them in one hand so that the four aglets are exposed but the rest of the shoelaces are hidden in your palm. Like this:
Have a friend select two of the aglets and tie those ends together (I recommend a square knot). Then, have your friend tie the other two ends together. Finally, offer your friend the following wager:
You give me $1 if you formed one large loop.
I’ll give you $1 if you didn’t.
Is it a fair bet?
Too easy? Then try this. Take your shoelaces and your friend’s shoelaces, fold them in half, and then expose the eight aglets. Choose two at a time and tie them together. The wager remains the same.
Now is it fair?
Is it possible to create a fair wager with any number of shoelaces? If so, how many?
I appreciate that the Super Bowl unites all Americans in their inability to read Roman numerals.
(Caution — oxymoron ahead!) If you’re a smart football fan, then you’ve invited folks to your house to watch the game, so you don’t have to drive home drunk on a cold Sunday night in February.
If this year’s Super Bowl is like last year’s, your guests will be bored by halftime, with the Seahawks leading 22‑0.
Or if deflated balls are used, the Patriots could be leading 38‑7 at the end of the third quarter, like when they played the Colts two weeks ago.
In either case, you’ll need something to keep your guests entertained between commercials and after the Katy Perry halftime show. I suggest the following problem.
Imagine that the NFL has eliminated divisions, and there are just two conferences with 16 teams each. To simplify things, every team plays the other 15 teams in their conference exactly once each. At the end of the regular season, what is probability that every team within a conference has a different record? (Assuming, of course, that each team is equally likely to win on any given Sunday, and assuming that ties are not possible.)
It’s not a trivial problem, so I’ll give you some time to think about it. The answer will be revealed on Monday, February 2, 2015; that is, after the game.