Say, I have a fair coin (consisting of $H$ and $T$) and I am asked to find out the probability that I get $H$ exactly $3$ times, from a total of $5$ tosses, let's name this event $A$.
I am aware that $P(A)= \binom{5}{3}(\frac{1}{2})^{3}(\frac{1}{2})^{2}$
However, in terms of the number of combinations $\binom{5}{3}$, I am still rather confused, as I have always looked at the binomial coefficient (re: combinatorics), in terms of the urn model. It states for $\binom{n}{k}$:
For $n$ balls in a urn, there are $k$ "drawings of the balls" (do not know if I am using the correct english terminology), where after each drawing the ball is not returned to the urn, and the order in which the ball drawings is not considered.
I understand this illustration above, but how can I "translate" it to the coin-tossing problem I mentioned at the beginning? I've spent the last hour attempting to do so, but to no avail. Ultimately, we are looking at $5$ "balls" $H, H, H, T, T$ and looking at the number of ways that sequence can be ordered. But this illustration just confuses me more.
Any explanation, as to how I can "connect" the urn model to the coin-tossing example?
Thank you.
Fate knows that you are going to do a coin tossing experiment and will toss the coin $5$ times. Fate decides to let you obtain $3$ heads. In order to distribute these $3$ heads randomly Fate puts $5$ balls numbered $1$–$5$ into a hat and randomly draws $3$ of them without replacement. The numbers on these balls will be the tosses in your experiment that come up heads.