The Wilson score interval is a confidence interval of the probability of success, based on the proportion of successes in a set of Bernoulli trials (a Bernoulli trial is a trial in which exactly two outcomes are possible: success or failure). The interval is given by the following formula:
The two values given by the formula are the upper and lower bounds of the interval. nS and nF are the number of successes and failures, respectively, and n is the total number of trials (equivalent to nS + nF). z is a parameter dependent on the level of confidence desired. For the purposes of this challenge, z = 1.96 will be used (corresponding to a 95% confidence interval)1.
Given non-negative integers nS and nF, output the bounds of the Wilson score interval.
- The outputs must be as accurate as possible to the true values, within the limits of your language's floating-point implementation, ignoring any potential issues due to floating-point arithmetic inaccuracies. If your language is capable of arbitrary-precision arithmetic, it must be at least as precise as IEEE 754 double-precision arithmetic.
- The inputs will be within the representable range for your language's native integer type, and the outputs will be within the representable range for your language's native floating-point type.
- n will always be positive.
- The order of the outputs does not matter.
n_s, n_f => lower, upper
0, 1 => 0.0, 0.7934567085261071 1, 0 => 0.20654329147389294, 1.0 1, 1 => 0.09452865480086611, 0.905471345199134 1, 10 => 0.016231752262825982, 0.3773646254862038 10, 1 => 0.6226353745137962, 0.9837682477371741 10, 90 => 0.05522854161313612, 0.1743673043676654 90, 10 => 0.8256326956323345, 0.9447714583868639 25, 75 => 0.17545094003724265, 0.3430464637007583 75, 25 => 0.6569535362992417, 0.8245490599627573 50, 50 => 0.40382982859014716, 0.5961701714098528 0, 100 => 0.0, 0.03699480747600191 100, 0 => 0.9630051925239981, 1.0
zvalue is the
1-α/2th quantile of the standard normal distribution, where
αis the significance level. If you want a 95% confidence interval, your significance level is
α=0.05, and the