If you flip a coin 8 times how many possible outcomes are there?

Answer 1

The following notes answer the above question completely and neatly. It's better in maths to try to keep it simple.

Looking at the different possible outcomes; let H = Head and T = Tail. Then the different possible outcomes are (8H +0T) meaning 8 Heads and Zero Tails, or (7H + 1T) or (6H + 2T) or (5H + 3T) or (4H + 4T) or (3H + 5T) or (2H + 6T) or (1H + 7T) or (0H + 8T) giving 9 different outcomes (possibilities).

The 9 different outcomes are not equally probable, (8H + 0T) and (0H + 8T) are the least likely while (4H + 4T) will be the most likely. However, totaling all possible outcomes there are 256.

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There are 256 possible "histories" of the whole session, corresponding to

the number of possible values of one binary 'byte', where each flip fills in

one bit of the byte.

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Another Answer:

As is usual in science, the answer depends on how you look at the problem, and on your definition of the terminology that you use. Nine different outcomes, as described above, is correct when you decide to consider only distinguishable or, more, correctly, combinatorial, outcomes.

However, in statistics and probability, we don't usually deal with combinations, we deal with permutations, because looking at permutations allows you to see the indistinguishable outcomes, outcomes which do, in fact, affect the probability of the result.

In this case, throwing a coin 8 times (or with identical results, throwing 8 coins one time), there are multiple permutations that yield the same combination. For instance, there are eight permutations of one coin being heads, there are 28 permutations of two coins being heads, and so on and so forth, leading up to a grand total of 256 different permutations of 8 coins, yielding "only 9 outcomes".

So, in summary, there are 9 distinguishable "outcomes", or combinations, while there are 256 indistinguishable "outcomes", or permutations.

(To clarify "indistinguishable", I should say that you could certainly mark each coin with a number, say from 1 to 8, and then you would be able to distinguish the outcomes or heads or tails as a function of which coin was thrown. Similarly, throwing only one coin at a time, you could write down the outcomes as they occur, and they would certainly be "distinguishable". In this context, "indistinguishable" simply means that, in the absence of some tracking system, a throw, for instance, of HTTTTTTT, or THTTTTTT, would not be different - they would both be counted as (1H + 7T), using the notation of the earlier answer.)

Now, to nail this down for completeness sake, because the answer does matter, even though the question did not ask it, take a look at probability...

Probability is simply the number of permutations of a desired result divided by the number of permutations of all results. Since there is one permutation of no heads (0H + 8T), the probability of no heads is 1 in 256, or about 0.00391. On the other hand, since there are 8 permutations of one head (1H + 7T), the probability of one head is 8 in 256, or about 0.0313. Going one more step, the probability of two heads is 28 in 256, or about 0.109.

Understanding the distinction between combinations and permutations, and distinguishable versus indistinguishable, is key to understanding probability.