Category Archives: Probability

In like a Lion; Out like a Lamb?

This morning as I was walking to work through whipping wind and freezing cold temps, I thought to myself “March, you’re supposed to be going out like a lamb.”  Goodness knows it sure came in like a lion!  Then, I started thinking about Groundhog’s Day and how unreliable that good old Phil actually is!  That made me wonder . . . is there really any truth to this whole lion/lamb thing?

I spent the better part of my morning trying to track down an answer!

This is what I did:

I’m only concerned with how March comes in and out, and not what happens in the middle of the month, so I thought I’d look at the first 7 days of March and the last 7 days of March.  Then, I did a quick search and found that the average March temperature in Iowa for the past 150 years is 34.5 degrees Fahrenheit.

Given this information I decided I would define “Lion” to be a 7-day span in which 4 or more of the days had an average temperature that was less than the average monthly temperature.  Then, a “Lamb” was a 7-day span in which 4 or more of the days had an average temperature that was greater than or equal to the average monthly temperature.

I obtained daily average temperature data for Des Moines from The University of Dayton Average Daily Temperature Archive dating back to 1995.  Because the month of March isn’t over yet for 2014, I didn’t use the temperature data for any of the days in March 2014.  This is what I came up with:

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In the past 19 years March has followed the “In like a Lion; Out like a Lamb,” pattern in 13 different years (or 68% of the time).  It has followed an “In like a Lamb; Out like a Lamb” pattern 5 different years (or 26% of the time).  And once in the last 19 years March has come in like a Lion and gone out like a Lion . . . according to the average daily temperature in Des Moines.

This got me thinking about a few things:

1. This saying seems to be a little more accurate then the Groundhog shadow thing.

2. There weren’t any “In like a Lamb; Out like a Lion” years . . . I wonder how many times that has happened in the past 150 years (if at all)?

3. What do you think of my definition for Lion-like weather and Lamb-like weather?  Would you define it another way?  If so, how?

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Don’t worry the groundhog isn’t right anyway . . .

This year my two oldest children really got in to the whole Groundhog Day thing . . . so much so that when they heard on the Today Show that Punxsutawney Phil saw his shadow, signaling 6 more weeks of winter the oldest insisted that there be a “redo,” because he was not going to endure 6 more weeks of winter.

See?  They still have fun in the winter!

See? They still have fun in the winter!

“You guys,” I said.  “Just relax.  The groundhog can’t really predict the weather.  It’s just something fun to do on February 2 every year.  It’s just folklore.”  (My son is currently learning about folk stories in 2nd grade, so I thought this might sell my explanation!)

“How do you know mom?”  (that was my kindergartener, not convinced and not happy about the groundhog’s prediction).  “Maybe he’s right and you’re just saying its folklore so we won’t be sad.”

I don’t really remember how the rest of the conversation went, mostly because there wasn’t any use in arguing with either of them and also because by then, they had completely lost interest in the conversation.  But that kindergartener, she got me thinking.  How often is the groundhog right?  Does he predict the end of winter enough that there might actually be something to this whole “groundhog sees his shadow” thing?

U.S.A. Today wrote an article about the groundhog’s predictions and the National Weather Center also has a thing or two to say about how accurate the groundhog actually is; they report the groundhog has been wrong 15 times and right 10 times.  But here’s the thing . . . according to both websites the groundhog either predicts 6 more weeks of winter or an “early spring.”

Who decides if spring has come early?  How do we know if winter has lasted for 6 weeks?  Is this measured by the air temperature?  The amount of snow on the ground?  The vernal equinox is the “official” start of spring, but clearly that’s not what people are talking about when they refer to the groundhog’s prediction . . . otherwise, no prediction would be needed!

I suggest we get a little more precise about what we mean by the groundhog’s prediction, before we decide whether his predictions are accurate or not.  I propose that if the temperature for half or more than half of 6 weeks after February 2 (so that’s the week of 2/2, 2/9, 2/16, 2/23, 3/2, 3/9, 3/16) has an air temperature at or below the national average for the last 100 years that counts as 6 more weeks of winter; otherwise, spring.  According to this definition of an “early spring” Phil has correctly predicted the weather for the next 6 weeks, 11 times over the course of the last 26 years.

But, I agree spring is much, much better!

But, I agree spring is much, much better!

If we recognize this as an example of binomial probably, we know that about 30% of the time we would have been able to be just as accurate as Phil by flipping a coin where “heads” is early spring and “tails” is 6 more weeks of winter.  I’d say that qualifies as a folktale to me, wouldn’t you?

Math Dice Game

My mom bought me this little game called “Math Dice” for Christmas this year.  Have you heard of it?  I hadn’t, and truth be told, I’m not sure my mom realized it was a game she was buying it for me.

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When I opened the package she said “I thought you could do something creative with those dice and your math blog.”  In the days after Christmas, I scooped the unopened box of dice into our “junk drawer” (sorry mom) and rediscovered them this weekend while cleaning.  On the back the of the box were the directions to the “Math Dice Game.”

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This morning I had a little extra time, so I thought I’d give the game try!

Step One: Roll the 12-sided target dice

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Step Two: Roll the three 6-sided scoring dice.  Combine the three scoring dice in anyway to match or come closest to the Target Number.

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Ummm, really?  1, 1, 2?  The closest I could get to 80 was 24.  This is what I did:

(1+1+2)! = 4*3*2*1.  Can you get closer?

My second roll of the Target Dice was 36

My Scoring Dice roll was 6, 4, 2

Super easy: (6*(4+2))=36.  Did you get 36 another way?

My last roll was 20

And my scoring dice were 6, 3, 1

I couldn’t get 20.  I could get 18 and 21, but 20 right on the money was a little tricky.  Can you do it?

Much, much more to come about this fun Math Game, with my new Math Dice . . . I’m working on a table of possible Target Number combinations as we speak!

M&M’s Revisited (for the Last Time!)

If you haven’t been here before, then you don’t know that we’ve already talked about M&M’s twice (here and here) and you don’t know that we’ve talked a little bit about the colors of M&M’s in the bags.

Well, today I want to keep talking about the different colors of M&M’s in the bags.  Except today I want to talk about the percent of M&M’s that are red, orange, yellow, green, blue, brown.  Before we continue our M&M discussion, do you have a guess?   That is, what percent of the M&M’s manufactured are red, orange, yellow, green, blue, brown?

Hmmm . . . Let’s pretend that we don’t know (maybe you really don’t!).  I think a pretty educated guess would be that 16.67% of the M&M’s are red, 16.67% of them are orange, 16.67% yellow, etc., etc.  Can you live with that guess?

I’m going to use the data I collected in my last M&M post, except instead of individual bags I’m going to look at my entire sample of M&M’s.

Here’s the percentage breakdown of M&M’s:

Screen Shot 2013-11-26 at 9.52.31 AM

Let’s make a nice table, based on what I would expect to get, given my educated guess of 16.67% of each color and what I actually got:

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So, I wonder if the distribution of colors I got in my sample would be likely, if the colors of M&M’s really were distributed evenly at the manufacturer?

Luckily for us there’s a statistical test we can use to answer that exact question.  And, luckily for us its a pretty straightforward test to understand!  It’s called the Chi-Square Goodness of Fit Test.  The Chi-Square Goodness of Fit test compares the observed values (in our case my M&M colors) to the expected values (if our initial assumption was true).  In our case we would subtract the expected value from the observed value and square the difference.  Then, we would divide by the expected value.  We’d do this for each color of M&M and add up the results.  Don’t worry, I’ll do it (actually, I did it with the help of this website). . .

Based on the Chi-Square Goodness of Fit Test it’s fairly reasonable to assume that I could have gotten this distribution of M&M colors given the fact that M&M Mars makes 16.67% of each color of M&M’s.

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So, here’s my next question?  Do they?

(So here’s the thing, about 5 years ago the M&M Mars website used to answer this exact question, but in 2008 they stopped.  This person wrote to M&M’s and posted the response)

Use the distribution for Milk Chocolate M&Ms detailed by M&M Mars and run another Chi Square Goodness of Fit Test with my data (or your own, if you collected any).  How does this compare to the 16.67% guess?

 

Reblogged from The Atlantic: The Myth of I’m Bad at Math

Today I’m reblogging a post I saw in The Atlantic this morning.  I’m reblogging it because I think the authors, Miles Kimball and Noah Smith, do a great job of articulating a sentiment that I think many, many mathematics teachers share.

(And because they talk about Terence Tao, and any true Math Warriors fan knows his importance in the world of mathematics!)

Enjoy.

 

 

This is the second post in a trilogy!  Read the first post here.  And when you’re done with this post, read the last post here.

Have you been counting chocolate?

Good!

Did you eat the chocolate you counted?

Even better!

If we pick up where we left off last week, you’ll recall the M and M debacle (the debacle being the fact that my children fought, not the fact that I fed them chocolate before 8am)!

You remember the fight, don’t you?  You know, the one where my daughter said it wasn’t fair that her brother got all the colors of the M&M’s in his bag and she didn’t?

Then, I asked you to try to figure out how likely that situation was to occur?  And then . . . I asked you to buy lots and lots of M&M’s?

Well, did you?  I did!

See . . .

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The question that led to all of this M&M consumption was “How likely is it that a bag of fun sized M&M’s will contain exactly two colors of M&M’s?”

I asked you to start by looking at a sample of M&Ms.

My sample contained 17 bags of fun sized M&M’s and 299 candies.    When collecting the data to answer this question, I did two things.

1) Counted the total number of M&M’s in the bag.

2) Counted the number of each color of M&M in the bag.

My piles looked like this:

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And the data I collected looked like this:

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If you look closely, you’ll see that many times during the data collection phase I opened a bag of M&Ms that was missing one color.  In fact, this particular event occurred approximately 35% of the time.

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And when a color was missing, 33.3% of the time it was the blue or brown M&Ms that were missing and 16.67% of the time it was the orange or yellow M&Ms that were missing.

In my study, two or more colors were missing 0% of the time.

Considering this data, it seems very, very unlikely that my daughter would have opened a bag of M&M’s containing only two colors of M&M’s.  In fact, in my data collection this occurred 0% of the time.

So, here’s my next question; since the instance my daughter described happened 0% of the time during my study does it mean it will never happen?  How do you know?

Could I use this data to make an educated guess that my daughter my have just been picking a fight with my son, and that her bag of M&M’s did, in fact, contain more than just two colors of M&M’s?

To be continued . . .

M and M’s

This actually ended up becoming a 3 part series . . . all devoted to M&M’s!.  This is the first post in the series, but you can read the other two here and here.

This morning a box of fun-sized M&M packages got delivered to our house (thanks to my mother-in-law aka Grandma).  Although this seems like a thoughtful gesture, its a major problem.  Why?  You might ask?  Easy.  I have absolutely no will power when it comes to M&M’s, so much so, that a package of M&M’s may have been opened at breakfast this morning.  And a package may have also been opened for my 7 year-old.  And my 5 year-old.  And my 2 year-old.  (Don’t be impressed with my husband, the only reason he didn’t open a package is because be was already at work!)

Chocolate in the morning.  Everyone should be in a good mood, right?  Even better is chocolate covered in brightly-colored candy coating.  Unless, of course, that candy coating happens to lay the groundwork for a fight between two pint-sized humans.

“Mom! Jack’s package of M&M’s has all the colors and mine only has two!  That’s not fair!  Tell him that’s not fair!’

Hmmm, I’m not sure that “not fair” is the phrase that should be used to describe this particular incident.  How about “Mom!  That’s not likely!”

I know, I know “not likely” is not nearly as moving as “not fair,” but this is a blog about mathematics after all.  Fairness is not something that I can speak to in this context, but likely, now that’s something we can talk about!

Here’s what I want you to do.  And you have to promise to not cheat.  (Promise).  Buy a few bags of M&M’s this weekend.  As many as you see fit and go ahead and try to determine how “likely” it is that someone will get all the colors of the M&M’s in one fun sized bag.  Also, see if you can’t figure out if M&M Mars makes equal numbers of red, orange, yellow, green, blue, and brown M&Ms.

Remember, you already promised you wouldn’t cheat.  So I’m trusting you not to just Google this as soon as you’re done here.  Besides, you’d get to eat the project when you’re done.  If you just Google it, there won’t be anything good to eat.

I’m going to answer this question too.  And I’ll show you how I went about answering it . . . next week.  In the meantime, send me your methods.  Let’s see what we can come up with!