Math of Jury Sizes

Question

If we go by the assumption that a Jury is a representation of the public at large, then is 12 people statistically signficant?

When doing any scientific survey or poll, a sample of 12 people would be laughable. Particularly if the results are close to 50-50.

Is there any math done on this?

For example. We might go by some axioms such as, we should only convict someone if 2/3 of the public think they are guilty.

Then with that we can calculate the probability that a jury of 12 will wrongly convict or wrongly release a suspect.

For example if the proportional of the public that think the man is guilty is $x$. Then the probability that the jury will convict is if we convict if 2/3 of the Jury (8 or more men) say they are guilty is:

$$P(convict) = x^{12} + 12 (1-x)x^{11} + 66 (1-x)^2 x^{10} + 220 (1-x)^3 x^9 + 594 (1-x)^4 x^8$$

So for example if 3/4 of the public think the man is guilty he should be convicted. But the jury will convict 88% of the time and release on 12% of the time.

With the assumptions here is my question:

1. Assume public is effectively infinite
   2. A man should be convicted if 2/3 of the public think he is guilty.
   3. Given a number of jurors $N$ and what proportional of jurors $f(N)$ should we convict a man so that there is a 95% agreement with the public and the jury when 3/4 of the public think the man is guilty.
   4. (Extra: Does this suggest an ideal jury size?)

Answer 1

Let's suppose you want people to be very unlikely to be found "guilty" when fewer than two-thirds of the public think they should be convicted, but very likely to be found "guilty" when three-quarters or more of the public think they should be convicted. This will lead to the need for large juries, for example

• if the proportion of the public who thinks an individual should be convicted is $0.665$ then the jury should say "guilty" with probability of $5\%$ or less
• if the proportion of the public who thinks an individual should be convicted is $0.75$ then the jury should say "guilty" with probability of $95\%$ or more
• the jury is a random sample of $n$ members of the public and at least $g$ of the jury have to vote "guilty" (without discussion) for "guilty" to be the collective verdict of the jury

then, using the binomial distribution, the following values of $n$ and $g$ will provide a satisfactory solution, and there will be more cases a jury sizes increase. The first case is optimal in the sense of a minimum jury size

  n   g     
 311 221
 314 223
 318 226
 321 228
 322 229
 324 230
 325 231
 326 232
 327 232
 328 233
 329 234
 330 234
 331 235
 332 236
 333 237
 334 237
 335 238
 336 239
 337 either 239 or 240
 338 240
 339 241
 340 either 241 or 242

I suspect that this is not the size of jury that you were thinking about. If not, then you need to adjust the criteria in the first two bullet points