I need help explaining this problem to a student. The picture of the problem and his solution is shown below in the image.

“Calculate the probability of tossing a coin 8 times and getting 4 heads.”

His reasoning was that since there’s a .5 chance of getting a heads on each toss, then there should be a .5 chance of getting 4 of the 8 tosses to land on heads. (Note he used the same reasoning to get .25 for #11.)

I tried to explain to him that it was a binomial experiment but that didn’t convince him of a flaw in his reasoning.

I then tried to show him an example of only 4 tosses, in which the question was what’s the probability of getting 2 heads. I showed him the entire sample space, shown in the image below. I then found the probability by looking at all the possible ways to get two heads, and dividing it buy the number of outcomes in the sample space. I then showed him that the binomial experiment formula got the same answer as my reasoning.

He countered that I was double (or triple) counting some combinations. His claim was, for example, that HHTT was the same as HTTH since order didn’t matter, and therefore the sample space was much smaller then I was making it. This implies, in his mind, that there are only four options.

I need your help explaining the flaw in his reasoning, because I not doing a good job of finding it or convincing him. Is there a better way for me to explain/teach this? (By the way, I love when students do this. I wish I had more discussions like this one, where students didn’t just accept my first response but really probed my reasoning.)

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Very common misconception. Confirm with him that his model predicts—in the two coin situation—that one head/one tail will occur just as often as two heads and as two tails. Then flip two coins many many times and keep track. HH and TT will come up about equally. HT will come up about twice as often as these two.

I find that appeal to logic isn’t really effective. Instead, we have to predict and face the consequences of our predictions.

Thanks Chistopher! I’ll definitely keep the “predict and face consequences” idea in mind. I need to have students do more that anyway.

Here’s what I would try: stick with 8 coins. But instead of just asking for the probability of getting 2, 4 or 6 heads, have him fill out what he believes to be the entire probability distributions (0 heads to 8 heads). Hopefully, you and he can agree that this covers the sample space and must sum to 1, thus his responses above must be unreasonable.

This is what I tried first and it convinced him. Thanks!

I would try appealing to families of four children. Your student should agree there is only one way to get 0 girls in such a family: BBBB. However, there are four ways to get 1 girl in such a family: GBBB, BGBB, BBGB, and BBBG. A family with an eldest girl is much different than a family with a youngest girl.

If this doesn’t convince him, I would run a physical set of trials or virtual simulation. The relative frequencies should show up pretty quickly. 50 trials of 4 four coin flips at a time should yield vastly different relative frequencies for 4 heads than 2 heads. If he’s still not convinced, do 100 trials.

Thanks Doug! We talked about the families of four children situation earlier in the unit, but I missed the opportunity to draw a parallel between it and the coins. John (see below) created a simulation that I will show him on Monday that had the results you mentioned.

The student is right that the order doesn’t matter… but the coin that comes up heads does. Imagine all the coins were a different color so that you can distinguish the coins after flipping. It should become more obvious how HHTT is not the same as HTTH if we assume we always list the results in the same color order.

Great suggestions above.

I made a simulation for you in Google Sheets — see if that helps. http://goo.gl/eWwRGl

Oops, below comment should have been in reply to you.

Awesome spreadsheet! Excel and spreadsheets are so universal. Fathom is great (and I use it with my students), but this is something they could do even after they no longer have their school issued laptop!

This is a mess of awesomeness! Thanks! How long did it take to make?

I dunno. Time flies when you’re having fun! (The core eight-coin simulation went pretty quick.. five or 10 minutes. Tinkering and tailoring and polishing to share with a general audience and adding the other cases took a little longer.)