Question

jurors are randomly selected from a population of 3million residents. Of these 3million​ residents, it is known that 47​%are of a minority race. Of the 13jurors​ selected, 2are minorities.​(a) What proportion of the jury described is from a minority​ race?​(b) If 13jurors are randomly selected from a population where 47​%are​ minorities, what is the probability that 2or fewer jurors will be​ minorities?​(c) What might the lawyer of a defendant from this minority race​ argue?

Answer 1

a) p=0.154

b) P(X≤2)=0.019

c) He can argue that the proportion of minority jurors is not representative of the proportion of that minority in the population.

Explanation:

a) The proportion can be calculated dividing the number of jurors that are from the minority race by the total number of jurors:

`p=X/N=2/13=0.154`

b) We can model this with a binomial random variable, with sample size n=13 and probability of success p=0.47.

The probability of k minority jurors in the sample is:

`P(x=k)=\dbinom{n}{k}p^k(1-p)^(n-k)=\dbinom{13}{k}\cdot0.47^k\cdot0.53^(13-k)`

We have to calculate the probability that 2 or less minority jurors. This can be calculated as:

`P(x\leq2)=P(x=0)+P(x=1)+P(x=2)\\\\\\P(x=0)=\dbinom{13}{0}\cdot0.47^(0)\cdot0.53^(13)=1\cdot1\cdot0.0003=0.000\\\\\\P(x=1)=\dbinom{13}{1}\cdot0.47^(1)\cdot0.53^(12)=13\cdot0.47\cdot0=0.003\\\\\\P(x=2)=\dbinom{13}{2}\cdot0.47^(2)\cdot0.53^(11)=78\cdot0.221\cdot0.001=0.016\\\\\\\\P(x\leq2)=0.000+0.003+0.016\\\\P(x\leq2)=0.019`

The probability that 2 or less minority jurors is 0.019.