High School

In a recent senatorial election, 46% of the voters in a certain district were registered Democrats, 38% were registered Republicans, and 16% were registered Independents. The incumbent Republican was reelected over his Democratic and Independent challengers. Exit polls showed that 61% of the registered Democrats voted for the incumbent, 88% of Republicans voted for the incumbent, and 51% of registered Independents voted for the incumbent. Assuming that the exit poll was accurate, what is the probability that a vote for the incumbent was cast by a registered Independent?

Answer :

By applying conditional probability and the exit poll data, we calculate that there is an 11.71% probability that a vote for the incumbent was cast by a registered independent.

To find the probability that a vote for the incumbent was cast by a registered independent, we need to apply the principles of conditional probability. According to the information given, we have 46% Democrats, 38% Republicans, and 16% Independents. Of these, 61% of Democrats, 88% of Republicans, and 51% of Independents voted for the incumbent. We calculate the overall vote for the incumbent from each group separately and then determine the proportion that was cast by independents.

  1. Calculate the percentage of votes for the incumbent from Democrats: 0.46 * 0.61.
  2. Calculate the percentage of votes for the incumbent from Republicans: 0.38 * 0.88.
  3. Calculate the percentage of votes for the incumbent from Independents: 0.16 * 0.51.
  4. Add these up to get the total percentage of incumbent votes.
  5. Divide the percentage of incumbent votes from Independents by the total percentage of incumbent votes.

Following the above steps:

  • Democrats for incumbent: 0.46 * 0.61 = 0.2806 (28.06%)
  • Republicans for incumbent: 0.38 * 0.88 = 0.3344 (33.44%)
  • Independents for incumbent: 0.16 * 0.51 = 0.0816 (8.16%)

Total votes for incumbent = 28.06% + 33.44% + 8.16% = 69.66%

Now, to find the probability that a vote for the incumbent was cast by an independent:

Probability = Votes from independents for incumbent / Total incumbent votes
= 0.0816 / 0.6966
= 0.1171 or 11.71%