Some probability questions that are bugging me

[JasonJo]

The first problem deals with criminal convictions:
It takes 9 out of 12 jury members to convict a defendent. 65% of all defendents are guilty. Jury members make their decisions independent of each other. Probability that a juror votes a guilty person, innocent is .2 whereas the probability that a juror votes an innocent person guilty is .1

Find the probability that the jury renders a correct decision.

- Now i see this problem as a problem with two cases:
(1) when the jury votes an innocent person, not guilty
(2) and when the jury votes a guilty person, guilty.

leading to two separate probability, and i believe the two cases are mutually exclusive.

also, i let X be a random variable with binomial distribution where X is the number of jurors who vote an innocent person not guilty.
^ i believe that is case 1

and i let Y be a similar random variable as X, where X is the number of jurors who vote a guilty person guilty.

but my gut feeling makes me believe that this does not work.

any helpful hints from you guys?

and here is the other problem:
a certain typing agency employs 2 typists. one typists averages 3 errors per article and another averages 4.2 errors per article. the probability of using either typist is equal. what is the approximate probability that my article will have no errors?

i know it deals with expected values and the Poisson distribution, but i don't know how to setup the E(g(X))

ie, I let X be a random variable with Poisson distribution where X is the number of errors per article, and i find the values for each typist when X is 0.

do i just take their respective probabilities of them making zero errors, say A and B, and then setting up E(X) = .5A + .5B?

thanks guys, I've just been really unsure about myself lately and i lost a lot of my confidence

 

[EnumaElish]

Make a 2x2 table. Columns are true states of the world. Left column denotes innocent (Prob = 0.35); right col. denotes guilty (prob = 0.65). Rows are the jury's decision. Top row denotes "guilty verdict." Bottom row shows "not guilty verdict." For an individual juror, the prob. of the SE box is 0.2 (guilty found innocent or "false negative") and the prob. of the NW box is 0.1 (innocent found guilty or "false positive"). When the question specifies "a guilty person" or "an innocent person" you are being told which of the 2 columns the world is in, and the answer is a conditional probability. E.g. for an individual juror the prob of committing a false positive error is 0.1/0.35.

Can you work your way from here on?