Hi I have the following question and I'm stuck at part (v).

Suppose a defendant is convicted if at least 10 of 12 members of a jury vote the defendent guilty.

Suppose that probability that an individual juror votes a guility person innocent is 20%, whereas the probability an indiviudal juror votes an innocent person guilty is 30%.

Suppose that all jurors reach their decisions independently.

(i) What is the distribution of the nuber of votes of guilty if the defendant is innocent?

(ii) What is the probability that an inocent defendent is convicted?

(iii) What is the disrtubition of the number of votes if the defendent is guilty?

(iv) What is the probability that a guilty defendent is convicted?

(v) Suppose that 10% of all defendants are guilty. What is the probability that a convicted defendent is actually innocent? What is the probability that a defenden who is not convicted is actually guilty?

My answers for (i) Was the binomial distribution

(ii) I used the Binomial distrbiution and got $\displaystyle {12 C 10} * 0.3^{10} * (0.7)^2 = 1.909*10^{-4}$

(ii) I put binomial distribution again

(ii) I got $\displaystyle {12 C 10} * (0.8)^{10} * (0.2)^2 = 0.2834 $

Are these correct, Im not sure about (v) should I use the Bayes Theorem?