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# Probability & Statistics/easy probabilty question

Question
Dear Clyde,

I play "Plants vs Zombies 2" everyday and every night it has a bonus stage. You have right to hit 3 of the 16 pinatas to find the one and only biggest prize. SO i have to find that 1 pinata in 3 hits. But i cant do it for 700 days now. 700x3=2,100 trials and no big prize. I was trying to calculate the probabilty of "achieving" this. Is my calculation below correct?

"Not finding" the big prize probability:

1st trial   2nd trial     3rd trial
Probabilities   15/16    x    14/15   x    13/14   = % 81,25

So its highly that you cant find it right? Should i add them instead of multiplying? I am not sure.

But anyways if until now i am correct my real question is that everyday i make this probability real. Which means its nearly impossible right?

i mean (0,8125)^700 right?

its a huge number. Or everyday is a new beginning and its not related to each other maybe. that's where i am lost.

Maybe its all about i experience a normal 81,25% high probability everyday independent of the previous days. But again i cant understand the concept. Is it not like rolling a dice 700 times and not have one certain number? Coz 3/16 is very near to 1/6. Thats why i am confused. Everyday is a new dice rolling. And everyday i cant find the right pinata say i cant roll 1 or 2 or 3 or 4 or 5 or 6.

IF this is statistically impossible i would believe game has a cheat on users. it does not want to give us the right pinata.

Your ideas are correct, which means it might sound impossible that someone can win. However, we should be aware -- this is all based on an assumption. It may not be true that the game is intended to be perfectly random, as you describe.

Your calculation of 0.8125 is correct.

The probability of not winning 700 days in a row is also correct -- every day is another factor of 0.8125. And in fact, if you do the computation, you get:

0.8125^75 ≈ 7.5 × 10^(-64)

That is very unlikely.

However, the conclusion you should draw is that the game does not necessarily work on a perfectly random sort of outcome. Electronic gambling games, for example, have very specific odds that they are required to maintain -- sometimes those odds are not based on a perfectly random distribution, but a skewed distribution that maintains the appropriate odds.

So if it is the game developer's intention, it may be that this is a normal outcome. Perhaps there is only one winner per day? Perhaps they intend the average person to win only once per year (which means you're only a little bit unlucky).

It may also be a bug, rather than an attempt to cheat users.

It is always best to assume the most likely outcome -- one that requires the fewest and most neutral assumptions. It is tempting, but illogical, to assume that because an unlikely event has been observed that the explanation for that event must be equally unlikely, complex, or outlandish. The simplest explanation is still the best.
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Probability & Statistics

Volunteer

#### Clyde Oliver

##### Expertise

I can answer all questions up to, and including, graduate level mathematics. I do not have expertise in statistics (I can answer questions about the mathematical foundations of statistics). I am very much proficient in probability. I am not inclined to answer questions that appear to be homework, nor questions that are not meaningful or advanced in any way.

##### Experience

I am a PhD educated mathematician working in research at a major university.

Organizations
AMS

Publications
Various research journals of mathematics. Various talks & presentations (some short, some long), about either interesting classical material or about research work.

Education/Credentials
BA mathematics & physics, PhD mathematics from a top 20 US school.

Awards and Honors
Various honors related to grades, various fellowships & scholarships, awards for contributions to mathematics and education at my schools, etc.

Past/Present Clients
In the past, and as my career progresses, I have worked and continue to work as an educator and mentor to students of varying age levels, skill levels, and educational levels.