## Related questions with answers

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.

Which probability is relevant for determining whether 40 first lines for Democrats is significantly high: the probability from part (b) or part (c)? Based on the relevant probability, is the result of 40 first lines for Democrats significantly high?

Solution

VerifiedResult part (b):

$P(X=40)\approx 0$

Result part (c)

$P(X\geq 41)\approx 0$

When checking for a significant results, you need to include the more extreme cases. Thus for the event of 40 top lines we also need to take into account the case that there are more than 40 top lines (since this is further from the expected value of 20.5 top lines), which is the probability $P(X\geq 41)\approx 0$ in part (c).

Since the probability is very small (less than 0.05), the event of 40 or more top lines is unlikely to occur by chance and thus the result of 40 top lines is significantly high.

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