The question requires that we use the binomial probability distribution
To do this, we will use the binomial formula
Where
In our case
For the first question,
We are told to find the probability that exactly 6 jurors are Asian American
In this case,
So applying the formula
Thus, the probability that exactly 6 jurors are Asian American =0.0340
For the second question
We are told to find the probability of 6 or fewer than 6 are Asian Americans
We will repeat this for when r=1, 2, 3, 4, 5 and 6
And then we will sum the values up
Then
The probability of 6 or fewer than 6 are Asian Americans = P(0)+P(1)+P(2)+P(3)+P(4)+P(5)+P(6)
Thus,
The probability of 6 or fewer than 6 are Asian Americans = 0+0+0.00002+0.00026+0.00182+0.00921+0.03403=0.04534
The probability of 6 or fewer than 6 is Asian Americans=0.04534
For the third part
The probability of more than 10 will be
The sum of the probabilities when r = 11 and r =12
Thus, the total probability = P(11) + P(12)= 0.14072 +0.03713 = 0.17785
Hence,
The probability of more than 10 Asian-Americans will be = 0.17785