Innocent until proven guilty? In Japanese criminal trials, about 95% of the defendants are found guilty. In the United States, about 60% of the defendants are found guilty in criminal trials. Suppose you are a news reporter following ten criminal trials.(a) If the trials were in Japan, what is the probability that all the defendants would be found guilty? What is this probability if the trials were in the United States? (Round your answers to three decimal places.)(b) Of the ten trials, what is the expected number of guilty verdicts in Japan? What is the expected number in the United Sates? What is the standard deviation in each case? (Round your answers to two decimal places.)Japan = guilty verdicts = guilty verdictsU. S. = guilty verdicts = guilty verdicts

ai) P(X=10) in Japan = 0.5988

aii) P(X=10) in the Unites States = 0.00605

bi) Expected value of guilty cases in Japan = 9.5 cases out of 10

Expected value of guilty cases in the United States = 6 cases out of 10

bii) Standard deviation for Japan = 0.689

Standard deviation for the United States = 1.549

Step-by-step explanation:

Probability of being found guilty in Japan = 95% = 0.95

Probability of being found guilty in the United States = 60% = 0.60

a) Probability that 10 court cases are all guilty in Japan is a binomial distribution problem.

Binomial distribution function is represented by

P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ

n = total number of sample spaces = number of criminals studied = 10

x = Number of successes required = number of guilty cases required = 10

p = probability of success = probability of being found guilty = 0.95

q = probability of failure = probability of not being found guilty = 1 - 0.95 = 0.05

P(X=10) = ¹⁰C₁₀ (0.95)¹⁰ (0.05)⁰= 0.5988

aii) Probability that 10 court cases are all guilty in the United States too is a binomial distribution problem.

n = total number of sample spaces = number of criminals studied = 10

x = Number of successes required = number of guilty cases required = 10

p = probability of success = probability of being found guilty = 0.60

q = probability of failure = probability of not being found guilty = 1 - 0.60 = 0.40

P(X=10) = ¹⁰C₁₀ (0.60)¹⁰ (0.40)⁰= 0.00605

bi) Expected value is given as E(X) = Σ xᵢpᵢ

Summation of all variables and their probabilities.

Expected value in Japan = mean = np = 0.95 × 10 = 9.5 guilty people out of 10

Expected value in the United States = np = 0.6 × 10 = 6 guilty people out of 10

bii) Standard deviation = √(variance)

Variance = np(1-p)

For Japan, Standard deviation = √[10×0.95×0.05] = 0.689

For the United States, Standard deviation =√(10×0.6×0.4) = 1.549