this problem is to write R codes.

Probability:Explore more properties of the binomial distribution.

R:Basic programming inR- including optimization and testing techniques.

Jury Duty:There is doubtless some truth to the old saw "a camel is a horse designed by a committee". However, in the this lab, we will explore the possibility that for certain decisions, a committee may demonstrate better judgement than the individuals who make up the committee. Suppose that you are writing a new constitution and must make a decision about the size of a jury (N) and the rule by which it comes to a verdict (conviction takes place whenkor more jurors vote to convict). In your country, a typical citizen will vote to convict an innocent defendant 25% of the time. The same typical citizen will vote to convict a guilty defendant 80% of the time. LetP(C I) denote the probability that ajuryconvicts an innocent defendant. LetP(C G) denote the probability that a jury votes to convict a guilty defendant. The question we want to answer is: what is the smallest jury size,N, so that there is akfor which

and

P(C |I)0.001

P(C |G)0.99

(You want to make the mistake of convicting an innocent defendant at most 1 time in 1000, and convict the guilty at a rate of 99%.)

Notice that makingklarger makes the probability of conviction in either case smaller; this will help the first inequality to be true and make it harder for the second inequality to be true.

Question: What is the smallest jury size,N, so that there is akfor which

P(C|I)0.001

and

P(C|G)0.99

(You want to make the mistake of convicting an innocent defendant at most 1 time in 1000, and convict the guilty at a rate of 99%.)

UseRto answer this question. find the value ofN(andk), as well as yourR-code.