For a particular device’s location, you may assume that the weather each day is an independent random event, and you are given the predicted probabilities of the four possible weather types. Each of the $4^ n$ possible weather reports for a device must be encoded as a unique sequence of bits, such that no sequence is a prefix of any other sequence (an important property, or else the server would not know when each sequence ends). The goal is to use an encoding that minimizes the expected number of transmitted bits.
The first line of input contains an integer $1 \le n \le 20$, the number of observations that go into each report. The second line contains four positive floating-point numbers, $p_{\text {sunny}}$, $p_{\text {cloudy}}$, $p_{\text {rainy}}$, and $p_{\text {frogs}}$, representing the respective weather probabilities. These probabilities have at most 6 digits after the decimal point and sum to 1.
Display the minimum expected number of bits in the encoding of a report, with an absolute or relative error of at most $10^{-4}$.
Sample Input 1 | Sample Output 1 |
---|---|
2 0.9 0.049999 0.05 0.000001 |
1.457510 |
Sample Input 2 | Sample Output 2 |
---|---|
20 0.25 0.25 0.25 0.25 |
40.000000 |