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What is the probability that 4 out of 181 and 55 out of 181 belong to the same distribution? Which distribution will give the best fit?

Assume you have, say, a manufacturing process from which you draw 181 samples. The percentage of failures is not known. If you get 4 failures out of the 181 samples at one time and 55 failures at another time, what is the probability that the process has not changed, i.e. it produces the same percentage of failures?

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    $\begingroup$ How do you define "belonging to the same distribution"? $\endgroup$
    – Tim
    Commented Nov 29, 2017 at 9:01
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    $\begingroup$ Assume you have, say, a manufacturing process from which you draw 181 samples. The percentage of failures is not known. If you get 4 failures out of the 181 samples at one time and 55 failures at another time, what is the probability that the process has not changed, i.e. it produces the same percentage of failures? $\endgroup$
    – Gunnar Evang
    Commented Nov 29, 2017 at 9:44
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    $\begingroup$ Please edit your question to add those details, it seems that you are asking about some kind of anomaly detection. The mo0re details you give us, the more likely you are to get a valuable answer. $\endgroup$
    – Tim
    Commented Nov 29, 2017 at 9:48

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You can present your data as a contingency table:

          A   B
fail      4  55
success 177 126

and then use a chisquare test: 

    Pearson's Chi-squared test with Yates' continuity correction

data:  mytab
X-squared = 50.624, df = 1, p-value = 1.119e-12

and the conclusion should be clear.

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