Should you buy vouchers now or wait until general sale begins to get the 2 legendary airdrops?

image take from PlayToEarn

As of creating this post, the phase 2 sale period will end in about 9 days.

In order to help players make their decisions in determining whether they should go spend money on vouchers and then spend $4 to buy a pack to qualify for a chance to get the two legendary airdrops at a 0.33% each pack.

The two legendary airdrops

The first airdrop is Baakjira, an extremely majestic sea creature with 16 HP, 2 speed, and skills: Void, Slow, Strengthen, and Heal (self). An absolute anti-magic unit with high sustain.

The second airdrop is Uriel the Purifier, which looks like an enhanced Shieldbearer. With a whopping 4 attack, 6 armor, 3 speed (relatively fast considering his hulking-looking armor), 12 health, and skills: Flying, Recharge, Heal (self), and Redemption. An absolute beast of a unit...

Analysis

Based on this site, players are "guaranteed to receive one of EACH card for every 300 packs purchased". My interpretation is you will receive 1 of each airdrop (1 Baakjira and 1 Uriel) for every 300 packs purchased with vouchers.

Currently, one voucher sells for $2.67, which means you will need: 300 * $2.67 = $801 for vouchers and $4 * 300 = $1,200 for the basic cost of 300 chaos legion packs. Hence, the total costs are $2,001. Most people believe that this price is worth it because it would account for future airdrops (all 13 airdrops except Dr. Blight).

Using an "apple to apple" comparison, in the context of general sale, $2,001 would give you a whopping 500 packs but the downside is that you forfeit the potential 2 legendary airdrops above. But if you think of it this way, these 500 packs would give you a higher chance to get future airdrops (all 11 airdrops remaining). While I am unsure what the airdrop chances are of the future airdrops (e.g., 0.33% per pack?), I would roughly know the probability of getting these 2 legendary airdrops (yep, both of them) from the packs themselves (these 2 cards can be found in packs after phase 2 ends).

Enter Binomial Distribution Model

For those of you who are not familiar with this model, it is basically a statistical tool to figure out the probability of an event that would occur given the number of trials. It only has two outcomes: success or failure. No need to delve into the formula as it has an online calculator to save time.

Before we input the numbers, we need to determine some facts. First, once phase 2 ends, there would be a total of 18 legendary units in Chaos Legion (besides Dr. Blight) available in chaos packs. This means that there is a 1/18 chance (5.556%) that the legendary would be the 2nd or 3rd legendary airdrop. The next step would be to determine the chance of getting a legendary. Based on Splintercards, the chance to obtain a legendary card with a potion is 1.536%.

Therefore, the probability is 5.556% * 1.536% (prob of getting the particular airdrop legendary from packs * prob of getting legendary) = 0.08534%. At this point in time, most of you may feel that 0.08534% is lower than the 0.33% per pack, but what if all of these packs were opened?

I would be using the 500 packs ($2000) for analysis since the hypothesis is trying to find out if spending $2001 (300 vouchers and 300 packs) to get 2 guaranteed legendary airdrops is more worth it than spending $2001 worth of general sale packs (no need vouchers, 500 packs, exclude costs of 500 legendary potions).

Workings

1. The probability of success in a single trial (getting one legendary airdrop from all chaos legendary units and the base chance of legendary)
2. Number of Trials (this represents the total number of cards flipped, NOT the number of packs opened) So, 500 packs = 2500 cards (hence, no. of trials is 2500)
3. Number of successes (this refers to one successful pull of one of the legendary airdrops)

note: 0.08534% = 0.0008534
The no. of times successfully pulling one of the stated legendary airdrops is once (hence, 1).

Interpreting Results

As we can see in the probability of "P(X = x)", means that the probability of getting exactly one of these airdrops is around 0.25263 = 25.263%. Interestingly, if we look at the cumulative probability of "P(X > x)", it means that there is about 62.9% of getting MORE than 1 legendary airdrop. Similarly, for "P(X >= x)", it means that there is about 88.16% of getting one OR more legendary airdrops.

These probabilities are true only IF ALL 500 packs are opened.

Now, what if we change up the numbers a bit, let's say: 300 packs?

P(X = x) = 35.59%
P(X > x) = 36.61%
P( X >= x) = 72.21%

Conclusion

As we can see from the above two examples of inputs, opening 500 packs without spending vouchers will give you 62.9% of getting MORE than 1 legendary airdrops while 300 packs with vouchers give you 36.61% of getting more than 1 legendary airdrops. I feel that this comparison is fair as the costs for both scenarios are worth $2,001. So, what do you think? I hope this analysis helped you in your decision-making in the next few days, before phase 2 ends. Let me know if there are any errors in my parts of calculation or logic. Please share this post with others, especially newcomers intending to spend their first investment in Chaos Legion.

Thanks for reading!

Remarks

I was inspired by "Luke Plays to Earn" and "Dwayne Cunningham" on Youtube to make this post :D Great people~