In the case of Castaneda vs Partida, it was noted that although 80% of the population in a Texas county was Mexican-American, only 39% of those summoned for grand juries were Mexican-American. Assume that 12 jurors are to be summoned from a population that 80% Mexican-American. What is the probability that exactly 7 of the 12 jurors are Mexican-American? a) Carefully explain how this situation results in a binomial distribution. b) Write out the Binomial Probability Function without simplifying for this situation. c) Calculate the probability, to four decimal places, of selecting exactly 7 Mexican-Americans when 12 jurors were selected at random from a population that is 80% Mexican-American.
So p must happen ten times so: p7, and (1−p)(1−p) must happen 7 times, so: (1−p)3(1−p)7. That way the final answer, in my mind, should be p7(1−p)3p7(1−p)3 But it is not.
Could someone help me?
Regina/Sally/Linda ~ Pick a name and keep it please!
Please also be patient for a tutor who might be well versed in statistics. Most of our math tutors aren't.
Who is Regina/Sally/Linda?
If you don't know, you're in bigger trouble than we thought. They are the names who've posted from the same internet address in Spokane.
Certainly! Let's break down the problem step by step.
a) This situation results in a binomial distribution because it meets all the criteria for a binomial experiment. A binomial experiment has the following characteristics:
1. The experiment consists of a fixed number of trials.
2. Each trial has only two possible outcomes, often referred to as "success" (Mexican-American) or "failure" (non-Mexican American).
3. The probability of success (p) remains the same for each trial.
4. The trials are independent of each other.
In this case, we have 12 trials (selecting 12 jurors), each trial has two possible outcomes (Mexican-American or non-Mexican American), the probability of selecting a Mexican-American juror is fixed at 80% (0.80), and we assume that the selection of each juror is independent of others.
b) The binomial probability function is given by the formula:
P(X=k) = C(n,k) * p^k * (1-p)^(n-k)
where:
- P(X=k) denotes the probability of exactly k successes (in this case, selecting exactly 7 Mexican-Americans)
- n is the number of trials (12 jurors)
- k is the number of successful trials (7 Mexican-Americans)
- p is the probability of success in a single trial (0.80)
- (1-p) is the probability of failure in a single trial (0.20)
- C(n,k) represents the binomial coefficient or the number of ways to choose k successes from n trials, which is calculated using the formula: C(n,k) = n! / (k!(n-k)!)
c) Now, let's calculate the probability of selecting exactly 7 Mexican-Americans:
P(X=7) = C(12,7) * (0.80)^7 * (0.20)^(12-7)
To calculate C(12,7):
C(12,7) = 12! / (7!(12-7)!) = 792
Substituting the values into the formula:
P(X=7) = 792 * (0.80)^7 * (0.20)^5
Calculating this expression will give you the desired probability.