In the screenshotted problem, screenshot 1 shows the solution 1 using combinatorics only. screenshot 2 shows the solution 2 using combinatorics + probability calculations (i.e. [4! / 1! 3!] * [1/2]^4 for case 4 in screenshot 2).
Both solutions work. But, this is a coin toss problem where all probabilities are 1/2. I’m assuming this lets solution 1 work since everything is over the same denominator and nothing is altered by the 1/2 numerator.
Is solution 1 only possible when probabilities of events are all equal like with coin tosses?
If a “given probability” problem like this comes up, but it’s with varying probabilities for each event within each case (i.e. 1/2, 2/3, 1/3 etc) is solution 2 always the best to go with because of everything above?