Sentenced (to probability questions) for life: 10 jurors

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Tascja

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Hello, I'm having some difficulty with one of my probability questions. I am really confused, any help would be appreciated. Thank you so much.

In a trial it takes at least 10 votes from a 12-member jury to convict a defendant. Suppose that the probability that a juror votes a guilty person innocent is 20 percent whereas the probability that a juror votes an innocent person guilty is 10 percent. Assume each juror acts independently and that 65 percent of defendants are guilty. Calculate the proportion of defendants that are convicted.

Juror votes a guilty person innocent = .20
Juror votes a guilty person guilty = .80
Juror votes an innocent person innocent = .90
Juror votes an innocent person guilty = .10
 
Re: Sentenced to probability questions for life ;)

In a trial it takes at least 10 votes from a 12-member jury to convict a defendant. Suppose that the probability that a juror votes a guilty person innocent is 20 percent whereas the probability that a juror votes an innocent person guilty is 10 percent. Assume each juror acts independently and that 65 percent of defendants are guilty. Calculate the proportion of defendants that are convicted.

Juror votes a guilty person innocent = .20
Juror votes a guilty person guilty = .80
Juror votes an innocent person innocent = .90
Juror votes an innocent person guilty = .10
Click to expand...

I think you’ll find a tree diagram will make things very clear in this problem. Start by drawing two branches, one labeled “G (.65)” and the other labeled “I (.35)”, representing the defendants who are guilty 65% of the time and innocent 35% of the time.

From each of these branches, draw two more branches, one for each jury decision: either convicted or not convicted. Put the appropriate probabilities (per binomial distribution calculations) on each branch.

(Edited for error correction. Thanks pka.)
 
Re: Sentenced to probability questions for life ;)

Why have you both ignored the fact that 10, 11, or 12 votes will convict?
The probability that a guilty person will be convicted is \(\displaystyle P(C|G)=\sum\limits_{k=10}^{12} {{{12} \choose k}\left({.8}\right)^k\left({.2} \right)^{12 - k} } = .855\)
 
Re: Sentenced to probability questions for life ;)

Good "catch", pka. I completely rewrote the problem in my head, ignoring the meaning/intent of the first sentence. :oops:

Thanks for the correction!
 
Re: Sentenced to probability questions for life ;)

pkb, thank you!,

so if my n=12
and x = 10,11,12

P(CG, x=10) = .2301
P(CG, x=11) = .3766
P(CG, x=12) = .2824

To find the proportion of defendants that are convicted, would I add these?
P(C) = .8891

The proportion of defendants convicted is 89%.

or

since:
Juror votes a guilty person innocent = .65 * .20 = .13
Juror votes a guilty person guilty = .65 * .80 = .52
Juror votes an innocent person innocent = .35 * .90 = .315
Juror votes an innocent person guilty = .35 * .10 = .035

P(G) = .52 + .035
=.555 = .56

q=1-p
q=1-.56
q=.44

would the .9 and the .1 you include in your equation in fact be p=.56 and q=.44

thus,

if my n=12
and x = 10,11,12

P(CG, x=10) = .0388
P(CG, x=11) = .0089
P(CG, x=12) = .000951

To find the proportion of defendants that are convicted, would I add these?
P(C) = .04865

The proportion of defendants convicted is 4.9%.
 
Re: Sentenced to probability questions for life ;)

In a trial it takes at least 10 votes from a 12-member jury to convict a defendant. Suppose that the probability that a juror votes a guilty person innocent is 20 percent whereas the probability that a juror votes an innocent person guilty is 10 percent. Assume each juror acts independently and that 65 percent of defendants are guilty. Calculate the proportion of defendants that are convicted.

Juror votes a guilty person innocent = .20
Juror votes a guilty person guilty = .80
Juror votes an innocent person innocent = .90
Juror votes an innocent person guilty = .10
Click to expand...

You can still use a tree diagram to help visualize this. Start by drawing two branches, one labeled “G (.65)” and the other labeled “I (.35)”, representing the defendants who are guilty 65% of the time and innocent 35% of the time.

From each of these branches, draw two more branches, one for each jury decision: either found guilty or not guilty. To determine the appropriate probabilities to place on these branches, use a binomial distribution (per pka, as shown above). Note, however, that the .9 and .1 probabilities that pka used in his equation should actually have been .8 and .2 for P(C|G). This results in P(C|G) = .558 . This naturally means that a “not convicted” result of a guilty person has a probability of P(NC|G) = 1 - .558 = .442 . Thus the final probabilities of these two branches of the probability tree would be .65 times each of the above numbers:

.65(.558) = .363 convicted
.65(.442) = .287 not convicted

These are the numbers for the guilty persons who stand trial. Repeat this process for the innocent people who stand trial.

The probability that pka gave us was for an innocent person to be found not guilty: P(NC|I) = .889 . Therefore, the probability that an innocent person will be convicted is P(C|I) = 1 - .889 = .111 . Thus the final probabilities of these two branches of the probability tree would be .35 times each of the above numbers:

.35(.889) = .311 not convicted
.35(.111) = .039 convicted

Add the “convicted” numbers from each group to find the proportion of all defendants who are convicted:

.363 + .039 = .402 40% of all defendants are convicted.

(Hopefully I got it right this time. :wink: )
 
Re: Sentenced to probability questions for life ;)

From each of these branches, draw two more branches, one for each jury decision: either found guilty or not guilty. To determine the appropriate probabilities to place on these branches, use a binomial distribution (per pka, as shown above). Note, however, that the .9 and .1 probabilities that pka used in his equation should actually have been .8 and .2 for P(C|G). This results in P(C|G) = .558 . This naturally means that a “not convicted” result of a guilty person has a probability of P(NC|G) = 1 - .558 = .442 . Thus the final probabilities of these two branches of the probability tree would be .65 times each of the above numbers:
Click to expand...

where do you get the 0.8 and 0.2 for P(C|G) and then the 0.558?
 
where do you get the 0.8 and 0.2 for P(C|G) and then the 0.558?
Click to expand...

Juror votes a guilty person innocent = .20
Juror votes a guilty person guilty = .80

.558 from binomial distribution calculations as shown by pka above.
 
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