Binomial distribution - the smallest suitable number

[ningmeng7998]

Each year, a detailed study is conducted on a random sample of 36 Lorenz birdwing butterflies in Town A. A Lorenz birdwing butterfly is considered to be very large if its wingspan is greater than 17.5 cm. The probability that the wingspan of any Lorenz birdwing butterfly in Town A is greater than 17.5 cm is 0.0527, correct to four decimal places. For random samples of 36 Lorenz birdwing butterflies in Town A, ? is the random variable that represents the number of butterflies that are very large.
The probability that ? or more butterflies, in a random sample of 36 Lorenz birdwing butterflies from Town A, are very large is less than 1%. Find the smallest value of ?, where ? is an integer.
Is it correct to think it as to find the smallest integer ? such that 36C? * 0.0527^k * (1-0.0527)^36-K < 1% ? If so, how should I calculate it?
Note: (36C? as in choose ? from 36 with replacement)

 

[Dr.Peterson]

You need a cumulative probability (k or more), not just P(X = k). I would be inclined to make a spreadsheet and look for the first line where P < 0.01.

Are you saying that your "36C?" is something different from the usual binomial coefficient? Is that something you have been taught, or did you invent it? I am not familiar with what you seem to be describing.

 

[ningmeng7998]

You need a cumulative probability (k or more), not just P(X = k). I would be inclined to make a spreadsheet and look for the first line where P < 0.01.

Are you saying that your "36C?" is something different from the usual binomial coefficient? Is that something you have been taught, or did you invent it? I am not familiar with what you seem to be describing.

Thanks.
Yes, 36Ck is the normal binomial coefficient. I wrote down this way like how I normally type on my calculator because I don't know how to type it here in this forum.

 

[Dr.Peterson]

Thanks.
Yes, 36Ck is the normal binomial coefficient. I wrote down this way like how I normally type on my calculator because I don't know how to type it here in this forum.

You can use the subscript button to enter ₃₆C_k if you wish, though most of us would probably recognize just 36Ck.

The reason I asked was not the notation, but the mention of "with replacement", which is not appropriate for combinations.

I just made the spreadsheet I suggested; if you choose to do this "manually", I suggest that you calculate P(X>=k) as 1 - P(X<k), because it will be easier to do. The number will be relatively small, and this gives you fewer cases to add up.