Counting number of possibilities on flipping two coins. Why don't we double count 2 tails(TT) or 2 heads(HH)? According to texts, on flipping 2 coins we get 4 outcomes:
{HH, TT, HT, TH}.
If I understand correctly, if we count {TH, HT} as the same thing we get 3 outcomes:
{HH, TT, HT} //which arn't equally likely
If this is the case then in the former, since we consider TH & HT as different then why don't we double count TT & HH?, thus having 6 outcomes.
Eg: TT(first coin(or toss) is tails and second coin(or toss) is tails)
    TT(second coin(or toss) is tails and first coin(or toss) is tails)
 A: 
TT(first coin(or toss) is tails and second coin(or toss) is tails) TT(second coin(or toss) is tails and first coin(or toss) is tails)

These are exactly the same event. If the first coin is tails and the second coin is tails, then the second coin is tails and the first coin is tails (and vice versa).
TH and HT are not the same event. One says the first coin is tails and the second coin is heads, while the other says the first coin is heads and the second coin is tails. It is helpful to imagine we flip the coins one at a time, or maybe we can tell the coins apart in some way. Then TH and HT do not look the same to us. But TT will always look the same.
A: TH and HT aren't the same events. They are the same result if you are only counting the number of heads or tails, but to figure out probabilities by counting, you generally want your event space to consist of events with equal probability. The four equal-probability events for flipping two fair coins are HH, TT, HT, and TH.
A: Lightly paint one coin green and the other red.   Then the outcomes are $\{\rm\color{green}H\color{red}H,\color{green}H\color{red}T,\color{green}T\color{red}H,\color{green}T\color{red}T\}$, and each outcome is equally weighted.   So the probability for two heads is $1/4$, the probability for two tails is $1/4$, and the probability for one head and one tail is $1/2$.
Why would colouring the coins change the weight of these events?
A: In the end of the day this is a physical, not a mathematical question. Mathematically we can consider a probabilistic model where $\{HT\}$ and $\{TH\}$ is the same event having probability $1/3$. By the way, similar probabilistic model is used in quantum Bose-Einstein statistics, so the idea is not completely wrong. But in our classical world the better model is to consider $\{HT\}$ and $\{TH\}$ as two distinct events, having same probabilities as $\{HH\}$ and $\{TT\}$; this model makes better predictions about coin flipping in our real world. What remains is to convince yourself that the first model is wrong and the second is correct, and make it your intuition; but the intuition may fail if you find yourself in a different world.
