1. ## Probability question.

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 ${12 C 10} * 0.3^{10} * (0.7)^2 = 1.909*10^{-4}$
(ii) I put binomial distribution again
(ii) I got ${12 C 10} * (0.8)^{10} * (0.2)^2 = 0.2834$

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

2. In both (i) & (ii) you need to find the probability of at least ten votes.
That is ten, eleven or twelve.

3. For (i), I would say specifically B ~ (12, 0.3)

For (ii), there is also the case you didn't consider, which include 11 voting guilty and all 12 voting guilty.

For (iii), I think you missed something. Is it that an innocent is voted guilty? Or a guilty defendent voted guilty?

Same thing as in part (ii)

(v)

Make a probability tree for this one.

First branching, guilty and non-guilty with probabilities 0.1 and 0.9 respectively.
Then, the second branching at guilty; voted guilty and non-guilty with probability you got from part (iv) (be careful when assigning the probability)
Lastly, the second branchin at non-guilty; voted quilty and non-guilty with probabiltity you got from part (ii) (be careful here too)

(a) Then, find the probablity following the non-guilty, voted guilty branch.
(b) Find the probability following the guilty, not voted guilty branch.

4. Thanks. For part(iii) It is guilty defendant being voted guilty.

Your correct I never counted 11, and 12.

I'll post my answers when I do them.

5. Originally Posted by Unknown008
For (i), I would say specifically B ~ (12, 0.3)

For (ii), there is also the case you didn't consider, which include 11 voting guilty and all 12 voting guilty.

For (iii), I think you missed something. Is it that an innocent is voted guilty? Or a guilty defendent voted guilty?
....
Ok for part (ii) i did:

P(X >= 10) = 1 - P(X <= 9)
= 1- 0.9998
= 0.002

and for (iv)

P(X >= 10) = 1 - P(X <= 9)
= 1 - 0.4417
= 0.5583

6. For part (ii), I get 0.00020637... = 0.0002 (or $2.06 \times 10^{-4}$)

Right, I got that too

Try the last part now.