Question
Dear Mitch,
What are the odds that fraternal triplets will be two boys and one girl, which is what we are, as opposed to any other combination?
For the second time this week, it's us, the 'Triplets T' (R, C, and J)
Oh, and thank you for answering our other question about how we can really know how far the sun is from the earth. With our parents' help, we really understood what you said!
R, C, and J T.,
Boston, MA
Answer
Dear R, C, and J,
Well, with this seemingly simple question you struck upon one of my all-time favorite areas of math (and there are so many different math topics that I love!). And one of the reasons I find the subject so interesting is that there are often complications that a lot of people don't think of right away, so you can hear all kinds of adult s give all sorts of different answers, even though, usually only one of them is correct! The thing is a lot of magic formulas have been devised over the years, and they usually lead pretty easily to the correct answer. BUT, the problem very often is that adults either don't remember the formulas, or – more commonly – select the wrong 'magic formula', and end up with the wrong 'magic answer'!
So, to get to your question, it falls in the area of math we call 'probability', which you may know from school, but, of course, it is a tiny bit more complicated than most of the probability that you are likely to learn before about halfway through middle school.
First, though, just for a change from the way I usually answer questions, I'll come right out with the answer before I explain how and why I got it.
O.K., the answer to the question about the chances of fraternal triplets being two boys and one girl is this: A 1/4 chance.
Why?
Because the possibilities were these:
- 3 boys
- 3 girls
- 1 boy, 2 girls, and ...
- 2 boys, 1 girl
So, since we see 4 different possible combinations, and you three are one of them (two boys and one girl), you are the wonderful living possible combination that is #4 above.
Now, to show you how this question can get more and more complicated, think about this: you asked about 'combinations'. In math, and in most subjects, the word combination implies (quietly hints) that the order doesn't matter. For example, if two of my friends are married to each other (for now, let's call them Lori and Alan) and they always come to parties together, and I'm about to invite them to a big party I'm having to celebrate something exciting, someone might say, "Oh you're inviting Lori and Alan, that's great, they are a great COMBO (COMBINATION) and make all parties more fun and more interesting!" Now, the person could have said the same thing, but instead of saying 'Lori and Alan' the person could have said 'Alan and Lori', right? And it would not have made any difference because they are the same combination, and it really doesn't matter which one walks through the door first, because as long as they both show up, the 'Lori and Alan' combo is the same thing as the Alan and Lori combo (the same 'combination').
BUT, there are times when order DOES matter. And in those situations the fancy math word is 'permutations'.
Have you ever been at a big fancy wedding where the photographer is really working so hard to do everything he can to get as many great pictures as he can, that he doesn't bother to stop himself when all his hard work is starting to make the party a drop less fun? I mean, while he is focusing and focusing and snapping and flashing, you and other the guests are missing out on chances to talk to friends and relatives you may not have seen in a long time?
Well, you know why this happens? It happens because a lot of times photographers are not satisfied to get every COMBINATION of people in a particular shot, but he or she really is trying to go for every PERMUTATION of the group. That means that after he or she decides that a picture of the bride and the groom with their parents is important, he tries to get a picture of that group in every arrangement he can think of... (Maybe it's nice, he thinks, to have the bride and groom in the front, because it's their big day... and maybe it would be nice if the elderly people (grandparents) were in the front because that way they can be sitting in chairs comfortably while the younger ones stand behind them... then he might notice that the big white gown of the bride looks best in the light on the left side of the picture, near the window... but then maybe it would be a good idea to have the groom on the left, since the bride's dress already makes her shine a lot... Why not have him get a bit of that extra sunlight that the bride doesn't really need as much... then, wouldn't it be interesting to have all the women on one side to show the generations of women, and the men on the other side, instead of all mixed together.... )
-- OH LORD, WE'RE MISSING THE DESSERTS, LET'S RUN!!!!!!!!!!
So, if you had asked me what the odds were that two boys and one girl would be born in the order that you three were born (there probably was an order even if the time difference between 1st, 2nd, and 3rd to come out into the world was only a matter of seconds), then we would have a slightly more complicated question because, for example, BOY, GIRL, BOY is a different permutation (but not a different combination) from GIRL, BOY, BOY. And I'll stop there.
But one last thing: for anyone older who might be reading this: Since so many students seem to confuse the formula for combinations with the one for permutations, mostly because they forget which one to use for a particular question, I've come up with a little trick that students have told me they find helpful:
If you take a careful look at the word "COMBINATION," you'll notice that it begins with the word "COMB", and if you take a careful look at the word "PERMUTATION," you'll notice that it begins with the word "PERM". It doesn't cost anything or take very long to comb your hair, so you probably don't care so much if it comes out in the right order. For COMB, ORDER DOESN'T MATTER.
But when someone goes to a hair salon to get what's called a "PERM", it takes time and money. From what I understand, a "PERM" is when they use some kind of chemicals or something like that to give a person's hair a new shape for a while. Well, since it takes some time and costs money, you want your hair to come out in the right ORDER. FOR PERMS, ORDER MATTERS. PERMUTATIONS, is when the order of the things or people in the question matter for the correct answer. Use the math for COMBINATIONS WHEN ORDER DOES NOT MATTER.
Hope this helps,
And Happy New Year!!
-- Mitch