Question

Politics The County Clerk in Essex, New Jersey, was accused of cheating by not using randomness in assigning the order in which candidates’ names appeared on voting ballots. Among 41 different ballots, Democrats were assigned the desirable first line 40 times. Assume that Democrats and Republicans are assigned the first line using a method of random selection so that they are equally likely to get that first line.

b. Find the probability of exactly 40 first lines for Democrats.

Accepted Answer

Step 1: Recognize that this is a binomial probability problem. The random variable here is the number of times Democrats are assigned the first line, and it follows a binomial distribution because there are a fixed number of trials (41 ballots), two possible outcomes (Democrats or Republicans getting the first line), and the probability of success (Democrats getting the first line) is constant at 0.5 for each trial. Step 2: Write down the formula for the binomial probability: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success. Here, n = 41, k = 40, and p = 0.5. Step 3: Calculate the binomial coefficient (n choose k), which is given by the formula: (n choose k) = n! / [k! * (n-k)!]. Substitute n = 41 and k = 40 into this formula. Step 4: Substitute the values of n, k, and p into the binomial probability formula. This will give you P(X = 40) = (41 choose 40) * $$(0.5)^40 * (0.5)^(41-40). $$Simplify the expression by combining the powers of 0.5. Step 5: Simplify the entire expression to find the probability. This involves calculating the binomial coefficient, raising 0.5 to the appropriate powers, and multiplying the results. The final value will represent the probability of exactly 40 first lines for Democrats.

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Key Concepts

Probability

Binomial Distribution

Randomness and Fairness