Calculating probability with children
February 6, 2006 9:22 PM Subscribe
Looking for ideas for teaching probability to kids.
I've a small group of math-loving fifth graders for whom I'm trying to extend my curriculum's rather lame probability unit, which primarily deals with coin tosses and dice. I'm trying to think of some interesting things that we could reasonably calculate probabilities for.
My initial and probably best idea was to examine the probabilities that a given 5th grader might actually become a professional basketball player. It wouldn't be that hard to find the number of (American) players there are, how many other (American) kids there are who would potentially compete for those positions, and what fraction of them might be qualified and inclined to do so (maybe a poll for this last data point).
But I'm guessing there's a better idea out there, and fabulous resources that will make this easy and fun to teach. Is there? Are there?
I've a small group of math-loving fifth graders for whom I'm trying to extend my curriculum's rather lame probability unit, which primarily deals with coin tosses and dice. I'm trying to think of some interesting things that we could reasonably calculate probabilities for.
My initial and probably best idea was to examine the probabilities that a given 5th grader might actually become a professional basketball player. It wouldn't be that hard to find the number of (American) players there are, how many other (American) kids there are who would potentially compete for those positions, and what fraction of them might be qualified and inclined to do so (maybe a poll for this last data point).
But I'm guessing there's a better idea out there, and fabulous resources that will make this easy and fun to teach. Is there? Are there?
I'm not so fond of your basketball player idea because even a little kid isn't going to accept the idea that the pros are randomly chosen from the applicant pool. (It might still be interesting to compare "How hard is it to become a basketball player, assuming randomness? How hard to become a surgeon, assuming randomness?")
Definitely you should work on them on the Monty Hall problem.
Lotteries might be good. Lots of complicated counting there (maybe more than you want). Also the question of how how likely it is that two kids in the class have the same birthday, for different numbers of kids.
posted by Aknaton at 9:44 PM on February 6, 2006
Definitely you should work on them on the Monty Hall problem.
Lotteries might be good. Lots of complicated counting there (maybe more than you want). Also the question of how how likely it is that two kids in the class have the same birthday, for different numbers of kids.
posted by Aknaton at 9:44 PM on February 6, 2006
Teach them the rules of BlackJack and someday when they take a trip to Vegas they will remember their 5th grade teacher with love and appreciation.
It may sound a bit sinfull and I doubt BlackJack lessons will get the approval of the Principal, but really the laws of probability and trying to predict them with an educated guess is the eccence of gambling.
Pick any game that involves numbers and probability BUT don't ever tell the class that they have been actually participating in an experiment until after you get some results to equate. I bet you could find something fun at the TOY-R-US and use it for many years to come.
Games are so much fun in a class... and I believe that would be a lesson they would remember for life.
posted by SwingingJohnson1968 at 10:17 PM on February 6, 2006
It may sound a bit sinfull and I doubt BlackJack lessons will get the approval of the Principal, but really the laws of probability and trying to predict them with an educated guess is the eccence of gambling.
Pick any game that involves numbers and probability BUT don't ever tell the class that they have been actually participating in an experiment until after you get some results to equate. I bet you could find something fun at the TOY-R-US and use it for many years to come.
Games are so much fun in a class... and I believe that would be a lesson they would remember for life.
posted by SwingingJohnson1968 at 10:17 PM on February 6, 2006
I'm not sure if this is 5th grade level or not, but the Birthday Problem could be pretty fun.
It teaches some of the basics like: P(event happens) = 1-P(event doesn't happen), as well as the counting principle, and it has a result that's not very intuitive (in a class of 23 people there's a 50% chance to of them share the same birthday)
posted by chndrcks at 10:35 PM on February 6, 2006
It teaches some of the basics like: P(event happens) = 1-P(event doesn't happen), as well as the counting principle, and it has a result that's not very intuitive (in a class of 23 people there's a 50% chance to of them share the same birthday)
posted by chndrcks at 10:35 PM on February 6, 2006
I wouldn't write off the dice so quickly. I still remember very vividly the evening my father and I sat down with some dice and a piece of graph paper, and rolled ourselves a bell curve. Actually handling the dice and creating the graph was far more powerful than any abstract problem could have been.
I also like Rothko's idea on picking M&Ms. Probability is a very airy concept -- anything you can do to make it physical and prove your results on the spot.
By the way, I would NOT attempt to bring the Monty Hall problem into the situation at this age. Although the underlying logic is clean, the against-common-sense answer will likely alienate kids who are feeling they're getting the hang of probability.
If you feel the need to mess with them, you can always go with the "7 red socks, 5 blue socks in a drawer in a dark room -- how many do I need to pull out to make sure I have a matched pair" gag.
posted by tkolar at 10:38 PM on February 6, 2006
I also like Rothko's idea on picking M&Ms. Probability is a very airy concept -- anything you can do to make it physical and prove your results on the spot.
By the way, I would NOT attempt to bring the Monty Hall problem into the situation at this age. Although the underlying logic is clean, the against-common-sense answer will likely alienate kids who are feeling they're getting the hang of probability.
If you feel the need to mess with them, you can always go with the "7 red socks, 5 blue socks in a drawer in a dark room -- how many do I need to pull out to make sure I have a matched pair" gag.
posted by tkolar at 10:38 PM on February 6, 2006
I loved the Monty Hall problem when it was first explained to me, especially when it was broken down into coin flips (wikipedia calls this "increasing the number of doors").
The rules of roulette are pretty fun, too. Very few people realize that betting on black or red is not a true 50-50 bet -- they forget about 0 and 00 on the wheel.
posted by frogan at 10:55 PM on February 6, 2006
The rules of roulette are pretty fun, too. Very few people realize that betting on black or red is not a true 50-50 bet -- they forget about 0 and 00 on the wheel.
posted by frogan at 10:55 PM on February 6, 2006
My 4th grade teacher taught probability by playing "21" (he couldn't use the term Blackjack) with the whole class by going desk to desk and smoking all of us. He then went on to explain why he was able to beat almost all of us. It brought a lot of fun to the class, because we got a chance to best the teacher.
posted by Mijo Bijo at 11:57 PM on February 6, 2006
posted by Mijo Bijo at 11:57 PM on February 6, 2006
In these days of gambling, "21" may be a hard sell for approval, but it certainly is an excellent game for this type of thing. However, being 5th graders, I'd start with something a bit less then 1 in 52 chance of a specific thing being drawn.
Why not fill a Jar up with 2, 3, or 4 colors of marbles, and work out the probability of them drawing a specific color. Then work on multiple draws, like what are the chances of 4 draws in a row being red ones, or only drawing two of the colors instead of the other two.
Or you could go with candy instead of marbles, and at the end of the class, let them each have a piece. ;) I'm sure that would go over well. Maybe use Starburst or something. M&M's are too small. You could use that later. Just don't be totally random like the chances of drawing a blue one from a pack. Do something that you know the total volume of.
And definatly go with the dice thing. Sure, it's craps, but hey, dice are used in Monopoly and a ton of other board games that kids use, so you can focus on that aspect instead of gambling.
I work at a casino, and am very familiar with probability and statistics. My sister also teaches middle school science. If you like, I can forward the question to her and see what she thinks. My mother also used to teach Science, but has been doing college level stuff for years now.
posted by Phynix at 1:07 AM on February 7, 2006
Why not fill a Jar up with 2, 3, or 4 colors of marbles, and work out the probability of them drawing a specific color. Then work on multiple draws, like what are the chances of 4 draws in a row being red ones, or only drawing two of the colors instead of the other two.
Or you could go with candy instead of marbles, and at the end of the class, let them each have a piece. ;) I'm sure that would go over well. Maybe use Starburst or something. M&M's are too small. You could use that later. Just don't be totally random like the chances of drawing a blue one from a pack. Do something that you know the total volume of.
And definatly go with the dice thing. Sure, it's craps, but hey, dice are used in Monopoly and a ton of other board games that kids use, so you can focus on that aspect instead of gambling.
I work at a casino, and am very familiar with probability and statistics. My sister also teaches middle school science. If you like, I can forward the question to her and see what she thinks. My mother also used to teach Science, but has been doing college level stuff for years now.
posted by Phynix at 1:07 AM on February 7, 2006
Put it in the context of something they enjoy. What is the probability that the first video played after 4 pm on Tuesday will be by Britney Spears? by a non-American group?
Song xxx is at number 9 this week, and was number 15 last week. What are the chances that it will be in the top 5 next week?
posted by megatherium at 5:28 AM on February 7, 2006
Song xxx is at number 9 this week, and was number 15 last week. What are the chances that it will be in the top 5 next week?
posted by megatherium at 5:28 AM on February 7, 2006
My sixth grade math teacher used backgammon to teach us probability, and it made me a life long backgammon lover, as well as a decent player. We had a tournament and everything. I loved it!
posted by OmieWise at 6:18 AM on February 7, 2006
posted by OmieWise at 6:18 AM on February 7, 2006
A nifty thing that not many people understand is how sharp the normal distribution curve is. This leads to people trying to draw inside straights or trying to make a 9 in craps or other such foolishness.
It's sort of an illusion of choice. Set up some sort of random probability experiment. Running dice totals is one. Beforehand get the kids to specify all the possible outcomes of the probability event. Don't be too enthusiastic, because with even a small number of rolls you can end up with a very large number of states. Now most kids will be overwhelmed by the choice, seeing the large number of possible states.
When you start the event, even after only a few rolls or coin flips or whatever, you'll converge on the average very quickly. This is especially effective when you repeat a few times, or continue over a long period (rolling each morning, for example). What you're trying to do is impart how strong the central tendency is here.
If you're clever, you can plot the individual results over time and generate a normal curve. If you use something like a control chart of your results, you can graphically introduce your kids to the ideas of means, the normal distribution "bell" curve, even the standard deviation, all without algebra.
You can then get into the idea that probability is the relative areas under the normal curve. A way to demonstrate this concretely is to draw the curve on heavy paper, then cut out the areas and weight them. Using cut-and-weigh, your students can figure out rather complex probablilities by simply counting microstates then cutting out the appropriate areas of their normal curves (You'll probably want to graph out a big bell curve for them then provide photocopies---it's not easy for them to construct one on their own).
I'll say again, you don't need to do any algebra or even much math to get through this. You might want to look at teaching them to compute a mean, but I'd keep this for a later lesson when they've understood it intuitively.
posted by bonehead at 6:37 AM on February 7, 2006
It's sort of an illusion of choice. Set up some sort of random probability experiment. Running dice totals is one. Beforehand get the kids to specify all the possible outcomes of the probability event. Don't be too enthusiastic, because with even a small number of rolls you can end up with a very large number of states. Now most kids will be overwhelmed by the choice, seeing the large number of possible states.
When you start the event, even after only a few rolls or coin flips or whatever, you'll converge on the average very quickly. This is especially effective when you repeat a few times, or continue over a long period (rolling each morning, for example). What you're trying to do is impart how strong the central tendency is here.
If you're clever, you can plot the individual results over time and generate a normal curve. If you use something like a control chart of your results, you can graphically introduce your kids to the ideas of means, the normal distribution "bell" curve, even the standard deviation, all without algebra.
You can then get into the idea that probability is the relative areas under the normal curve. A way to demonstrate this concretely is to draw the curve on heavy paper, then cut out the areas and weight them. Using cut-and-weigh, your students can figure out rather complex probablilities by simply counting microstates then cutting out the appropriate areas of their normal curves (You'll probably want to graph out a big bell curve for them then provide photocopies---it's not easy for them to construct one on their own).
I'll say again, you don't need to do any algebra or even much math to get through this. You might want to look at teaching them to compute a mean, but I'd keep this for a later lesson when they've understood it intuitively.
posted by bonehead at 6:37 AM on February 7, 2006
My Finite Math teacher in high school taught us to play Ploy-Ploy. The entire game was: deal five cards, look at your hand, and think, "Hey, what are the odds of that?" We also had questions like "If there are 3 boys and 3 girls sitting in a (line/circle), what are the odds that Marten and Ilana are sitting next to each other?"
I'll try to think of his other examples. He was a good prof, he made it fun.
posted by heatherann at 7:02 AM on February 7, 2006
I'll try to think of his other examples. He was a good prof, he made it fun.
posted by heatherann at 7:02 AM on February 7, 2006
The Monty Hall problem is great and all, but I wouldn't start with it. Start simple.
My math teacher brought a mini-roulette along with some chips to class. We were all gathered around one big table, happily gambling along. Ah, the good times!
posted by cerbous at 7:05 AM on February 7, 2006
My math teacher brought a mini-roulette along with some chips to class. We were all gathered around one big table, happily gambling along. Ah, the good times!
posted by cerbous at 7:05 AM on February 7, 2006
One year for the science fair, my brother wanted to figure out the science behind the Price is Right game Plinko. He got a large peg board, put pegs in some holes, and some flat discs to drop down the board and went at it. He ended up with a bell curve, a useful concept to understand, as tkolar and bonehead have both mentioned. I think he was around 5th grade at the time, and he had tons of fun doing it, and the result was something visual and understandable. Perhaps you could make a similar game for the class. (You can ask simple probability questions, for each one they get right, they get a ball/disc to drop, and they get a special prize if they get it in the middle or something). Here's a pdf of a similar project somebody else did a with marbles.
posted by sarahnade at 9:25 AM on February 7, 2006
posted by sarahnade at 9:25 AM on February 7, 2006
When I was a kid, I had a copy of The I Hate Mathematics Book by Marilyn Burns. It has a chapter called "The Art of Probably" that surely has some fun ideas and concepts. (I can't recall exactly what it covers, but I will never forget that the difference between a permutation and a combination can be explained with scoops on an ice cream cone.)
The chapter titles are "1. Starting out right; 2. Street maths; 3. Maybe grownups aren’t as smart as you think; 4. Things to do when you have the flu; 5. A mathe-magic show; 6. How to always be a winner; 7. How many sides does a banana have?; 8. The art of probably. "
P.S. While the title of the book suggests it's for math-haters, kids who enjoy math will get even more out of it. It's really fun.
posted by hsoltz at 9:58 AM on February 7, 2006 [1 favorite]
The chapter titles are "1. Starting out right; 2. Street maths; 3. Maybe grownups aren’t as smart as you think; 4. Things to do when you have the flu; 5. A mathe-magic show; 6. How to always be a winner; 7. How many sides does a banana have?; 8. The art of probably. "
P.S. While the title of the book suggests it's for math-haters, kids who enjoy math will get even more out of it. It's really fun.
posted by hsoltz at 9:58 AM on February 7, 2006 [1 favorite]
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posted by Rothko at 9:33 PM on February 6, 2006