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.)