Combinations and Permutations in coin tossing

Question

I understand the formulae for combinations and permutations and that for the binomial distribution. However, I'm confused about their application to coin tossing.

Consider three tosses. Outcomes with two heads are HHT, HTH and THH. So, there are three and that's what you get if you use the formula for combinations.

With combinations "the order is not important". How does that apply in this case? In what way are these three outcomes different if the order is not important?

What are the 6 permutations?

Answer 1

Imagine that you have three balls numbered $1$, $2$, $3$ representing the flips on which heads may come up. (I'm using your scenario of three flips). You want to look at cases where you get two heads in three flips, so you are going to pick two of the balls. Picking first ball $1$ and then ball $3$ gives the same result (sequence of tosses) as first picking ball $3$ and then ball $1$ (both result in the outcome HTH).

Thus you are making an unordered selection of two of the three positions.

Answer 2

The six permutations exist only when you have three possible outcomes, not with the coin toss. If you have outcomes A, B and C, then the permutations are: ABC, ACB, BAC, BCA, CAB and CBA.

Answer 3

Permutations count how many ways you can order $n$ distinct objects. So if you had a 3 sided object you tossed you'd get $3!$ patterns.

Here there are only 2 outcomes. So if you now label the 3 tosses as 1st, 2nd and 3rd, there are $2^3$ possible toss patterns of 3 tosses.

If you look at a restricted case, such as 2 heads one tail, then you can choose the positions of the 2 heads in $\binom{3}{2}$ ways. Which is the same as choosing the position of the single tail in $\binom{3}{1}$ ways. This is why your event "2 heads one tail" can happen in the number of ways you pointed out, given by the formula for combinations.

Answer 4

I marked paw88789's response as the answer because it helped me most among those here, but I did look in lots of other places for an explanation. I found a helpful one here:

http://people.reed.edu/~jones/Courses/P03.pdf

It suggests thinking of 5 distinguishable coins and considering the number of ways you can arrange H1, H2, H2, T1, T2. This is 5!. Since there are in fact three indistinguishable heads and two indistinguishable tails the answer is 5!/(3!.2!).

It was the notion that order didn't matter that confused me, which might lead someone to think that HHT is the same as HTH is the same as THH, but as paw88789 says it's the order in which the heads are allocated to the positions heads occupy that doesn't matter.

Answer 5

Coin toss series can be viewed, depending on what you want to know, as either "combination" or "permutation" but in all cases "with repetition" (meaning same side can occur again and again).

*Probably the best page that summarizes the Combination vs Premutation with or without Repetition https://www.mathsisfun.com/combinatorics/combinations-permutations.html *

If the question is "How many ways a series of R coin tosses (N=2 sides) can go? Of these, how many will have 2 Heads in the row?", you are looking for Permutation with Repetition where "HHT" is different outcome from "THH".

Permutation with Repetition is the simplest of them all:

N to the power of R.

Example: 3 tosses of 2-sided coin is 2 to power of 3 or 8 Permutations possible.

In these, "at-least-2 Heads in a row" permutations are: HHH, HHT, THH - 3. Probability of "at least 2 heads in a row" is 3/8th (0.375)

If the question is "If you throw a 2-sided coin (N=2), R times, how many times can you get at least 2 heads?", you are looking for Combination (order is not important) with Repetition where "HHT" and "THH" are same outcomes (combination).

Combination with Repetition formula is the most complicated (and annoying to remember): (R+N-1)! / R!(N-1)!

For 3 2-sided coin tosses (R=3, N=2), Combination with Repetition: (3+2-1)! / 3!(2-1)! = 24 / 6 = 4

These are (because order is not important): HHH, HHT, HTT, TTT

If you are looking for "at least 2 Heads", 2 options match: HHH and HHT (order not important).

Probability of you getting at least 2 heads is 2 outcomes / 4 Combinations (with Repetition) = 0.5. Right?

Answer 6

I had the same problem and realised that, when the events are related to time, using permutations instead of combinations doesn't make sense, because event B cannot happen before event A, so we cannot choose different arrangements (permutations) of time-related events, so we must use combinations.

In this coin toss example, when we say "3 choose 2", it helps to think of the three events as not yet having a result. Only after we choose two of the three events will we give the events their heads/tails results.

So doing 3C2 we get {(Event 1, Event 2), (Event 1, Event 3), (Event 2, Event 3)}. Then we assign heads to the chosen events.

As you can see, we would be lying about the nature of the events, if we used permutations here, as Event 2 cannot come before Event 1. Therefore, we use combinations, which disregard order, to say what we really mean, which is that there can only be a single order.