I have already opened 26 Dragonite Eggs and have only got 5 out of the 6 basic Dragonite troops. This is statistically impossible and very annoying. How many Dragonite Eggs do I need to buy with Gems so I can get the 6th troop?
No, it is not.
If we assume the easiest case for now, that you pulled all five other dragons in a row at the beginning, the odds to miss the sixth one in 21 tries would be 0,83^21 = 0,021.
That’s about 1 in 50 players, who need at least 27 attempts. That is not even close to “impossible”.
For comparison, having three pieces of the same kind drop after matching three has a chance of 2,0%, so roughly the same as your bad dragon luck, while the famous Rising Shadows double kill (kill one troop, take two more down through trait) has a chance of 0,5%, a quarter of that.
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That’s the nature of gambling mechanics.
My Rising Shadows gets triple kills pretty often. What are the odds of that? (Not offering an argument, just curious 
)
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I’m sure you are killing a lot of troops as well. It’s a matter of perception, one does not open hundreds of dragon eggs every day. I’m sure, if you keep track, in the long run having your Rising Shadows kill two (or more) enemies at once after a kill will still be somewhere around a fourteenth of every fourteen times.
Here’s another small experiment, I suggest to the thread opener: Pick a physical die and roll it until every number has shown at least once. Count the number of attempts it took. Repeat this a couple of times, let’s say 5-10 rounds.
It’s the same random chance as with gem dragons, and you’ll be surprised how long a certain number can avoid you.
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One battle in 2915 (rounded down) will have Rising Shadows kill the three other troops when the first troop is killed. This ignores immunities.
Calculation: 1 / (0.07 * 0.07 * 0.07)
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Regarding the OP’s question about how many Dragon Egg they need to obtain all 6 troops, there is unfortunately no definite answer, only probabilistic ones.
This is known as the “coupon collector problem”. I’ll spare you the maths and give some results:
- Your are not alone: about 8% of the players need more than 26 Eggs to collect all 6 Dragons.
- About 54% of the players need 15 Eggs or less. This is the expected value.
- The expected value is not a certainty. The standard deviation is about 6 Eggs, which means 68% of players need 9 to 21 Eggs.
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Annoying? Yes.
“Statistically impossible”? (laughs in combinatorial probability spreadsheet)
Here’s how the overall probability of having Y unique dragons after X eggs actually maps out:
(Ask me how this is calculated. I dare you.)
15 eggs is the “expected value”, but as you can see there is a significant chance of NOT having a full set by then, and even a 3% chance of not even having 5 dragons yet!
Even by 30 eggs, there’s still a ~2.5% chance of NOT having the full set.
So it really is more likely than you think.
Alternative plots of the same data
Here’s a non-stacked version:
And here’s a Y-logarithmic plot:
Note how in a logarithmic plot, each series converges to a “straight line” aka. an exponential decay of * (# dragons / 6) per egg crafted, which is of course the basic probability of getting a duplicate.
(PS - if you are tracking your craft dates for individual eggs, or at least the sequential order, I’d love to add them to my spreadsheet data.)
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My suggestion would be this, every drop is a gamble with a percentage of troops that drop randomly. But I notice the names is in alphabetical order just like Monday - Sunday. Let’s say you have 28 eggs saved up each week you try your luck, if you tried it on Monday you might get Amethialas and if you use 28 eggs on Tuesday you might get Diamantina because 7 dragons with names that start with letters in order just like the 7 days of the week Amethialas=Monday, Diamantina=Tuesday, Emeraldrin=Wednesday, Garnetaertin=Thursday, Rubirath=Friday, Saturday=Sapphirax, Topasarth=Sunday or you can use 7 eggs for monday 7 for Tuesday 7 for Wednesday and so on. Just a suggestion. 
Me:
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