Answer to Question #127611 in Statistics and Probability for Harsha

Question

Computing Research team doing survey of raining in America.
City of London typically has rain on about 16% of days in April.
(a) What is the probability that it will rain on exactly 5 days in April? 15 days?
(b) What is the mean number of days with rain in April?
(c) What is the variance and standard deviation of the number of days with rain in April?
(d) How probability theories relate to Artificial intelligent programming?

Answer 1

The number of rainy days, $X$ can be represented by a binomial distribution with $n=30$ trials (the number of days in the month of April), success probability $p=0.16$ (representing a rainy day) and failure probability $q=1-p=1-0.16=0.84.$

Although it is not quite true that the weather (rain or not) is independent from day to day, we shall assume it is quite close to being independent, in order to use the binomial distribution.

$X\sim Bin(n, p)$

a) The probability that it will rain on exactly 5 days in April is

$$P(X=5)=\binom{30}{5}(0.16)^5 (0.84)^{30-5}\approx0.19116801$$

The probability that it will rain on exactly 15 days in April is

$$P(X=15)=\binom{30}{15}(0.16)^{15} (0.84)^{30-15}\approx0.00001308$$

(b)

$$\mu=np=30(0.16)=4.8$$

(c)

$$Variance:\sigma^2=npq=30(0.16)(0.84)=4.032$$

$$Standard\ deviation:\sigma=\sqrt{\sigma^2}=\sqrt{4.032}\approx2.008$$

(d) Probabilistic programming is an emerging field at the intersection of programming languages, probability theory, and artificial intelligence. 

In language translation, we want to translate a sentence S from source language to a sentence T in target language. We seek a perfect translation. 

We seek to minimize error by choosing those translations that have maximum probability of occurrence. Essentially, we eliminated hand-crafted rules and built a statistical model which uses training data on translations to learn good (i.e. most probable) translations.

Probability is fundamental in AI, as the statistical machine learning framework is based on probability of the input and output conditional distributions.