m05 questions 17 - someone please explain

OK, I’ll try.

Let’s begin with general case.

Imagine that a fair coin is tossed N times. Outcome of each toss is either tails or heads.
After N tosses we will get a sequence of heads and tails – something like that:

TTHHHHTTHTHH… etc.

Now the question is: What is the probability that from N tosses, there will be exactly K tails?

To answer this question, we need, first, to calculate the number of all possible outcomes of the experiment – namely, the number of different sequences of heads and tails. It’s easy to see that this number equals 2^N (we have N tosses, and each toss has two possible outcomes).

Next, we need to calculate how many of those sequences have exactly K number of tails – and the answer is C(K,N). I’ve omitted the reasoning behind the calculation here.

So, overall, the probability of getting exactly K tails from N tosses is C(K,N)/2^n.

Now, let’s return to our problem. Kate will have more than 10$ but less than 15$ if she wins 3 or 4 times. This means we need to calculate probability(3 tails from 5) = C(3,5)/(2^5) and probability(4 tails from 5) = C(4,5)/(2^5) and then sum these probabilities. And the solution provided with the problem does exactly this.

Well, this is it. I hope this explanation was helpful.