There are 2 available positions and 50 candidates, one half of whom are democrats and another half are republicans. If it was decided that the positions would be filled at random, then what is the probability that the both positions will be taken by members of just one party?

A. 1/25 B. 12/49 C. 1/4 D. 24/49 E. 1/2

(D) Let’s consider an outcome to be an ordered pair (x, y), where x is a member, who took position #1, y is a member, who took position #2. There are 50 × 49 possible outcomes.

There are 25 × 24 favorable outcomes for one party (x and y are from the same party). So there are 2 × (25 × 24) favorable outcomes in total.

The probability is (2 × 25 × 24) / (50 × 49) = 24/49. The correct answer is D. ---------- I don't understand the feedback.

If there are 50 total candidates (25 from each party), then the probability that someone fills position 1 from Party 1 is 1/2 (because it doesn't matter who fills the position as long as it is someone from Party 1) and then to fill position 2 the probability is 24/49. Combined 1/2 × 24/49 = 12/49.

If you relate this to the probability of flipping heads twice, it is 1/2 x 1/2 = 1/4 and 12/49 ~ 1/4

How is this scenario any different than flipping a coin twice other than you subtract one member who is already elected?

then the probability that someone fills position 1 from Party 1 is 1/2 (because it doesn't matter who fills the position as long as it is someone from Party 1)

Here is the moment in your reasoning that led to the wrong answer. You have calculated the probability for the specific party (Party 1). While the desired event is filling two seats with representatives from the same party. They both can be either from Party 1 or from Party 2.

Thus when you add the probability that the both seats go to Party 2 to your answer, you'll get the correct result: 12/49 + 12/49 = 24/49

Quote:

If you relate this to the probability of flipping heads twice, it is 1/2 x 1/2 = 1/4 and 12/49 ~ 1/4

Similarly, you should compare not to "flipping heads twice", but to "flipping one side twice". This side can be either heads or tails.

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