Suppose it can be assumed that 60 percent of adults favor capital punishment. Find the probability that on a jury of twelve members:
a) All will favor capital punishment.
b) There will be a tie.
To find the probabilities in this scenario, we can use the concept of binomial probability. The binomial distribution is used when there are only two outcomes: success and failure. In this case, success refers to a juror favoring capital punishment (with a probability of 60%), and failure refers to a juror not favoring capital punishment (with a probability of 40%).
a) To find the probability that all twelve members of the jury will favor capital punishment, we can calculate the probability of success (P) raised to the power of the number of trials (n), multiplied by the probability of failure (Q) raised to the power of the number of failures (which in this case is zero). Mathematically, it can be represented as:
P(all favor capital punishment) = P(success)^n
P(all favor capital punishment) = (0.60)^12
P(all favor capital punishment) ≈ 0.006
Therefore, the probability that all twelve members of the jury will favor capital punishment is approximately 0.006, or 0.6%.
b) To find the probability of a tie, we need to determine the number of ways the votes can be split evenly between the two options. In this case, we need exactly 6 jurors to favor capital punishment and the remaining 6 to not favor it. We can use the concept of combinations to calculate the number of ways we can choose 6 jurors out of the 12.
The formula for combinations is:
C(n, r) = n! / (r! * (n-r)!)
where n is the total number of items and r is the number of items chosen. In this case, n = 12 and r = 6.
C(12, 6) = 12! / (6! * (12-6)! )
C(12, 6) = 12! / (6! * 6!)
C(12, 6) = 924
Now, we need to find the probability of success (P) raised to the power of the number of successes and the probability of failure (Q) raised to the power of the number of failures:
P(tie) = P(success)^6 * P(failure)^6
P(tie) = (0.60)^6 * (0.40)^6
P(tie) ≈ 0.031
Therefore, the probability of a tie in the jury, where six members favor capital punishment and six members do not, is approximately 0.031, or 3.1%.