A poll is given, showing 40% are in favor of a new building project.

Question

A poll is given, showing 40% are in favor of a new building project.

If 5 people are chosen at random, what is the probability that greater than 3 of them favor the new building project?

Answer 1

There are two important facts in this problem to observe. Let me see if rewording them helps you solve this problem.

For each person polled, the chance that they favor the project is 40%.

We want to know if at least 3 of 5 will favor the project. This means we want to know the probability 3, 4, or 5 people out of 5 favor the project. This could also be phrased as 0, 1, or 2 people opposed to the project.

Can you break the problem down into something solvable now? Keep in mind that 40% is 2/5, so the chance of 3 or more people favoring the project may be relatively low. Use that to check that your answer makes sense when you find it.

Answer 2

Based on the poll, each person has a 40% or 0.4 chance of being in favor of the new building project. If you consider the case of selecting 5 people where the first 3 people are in favor, and the last 2 are not, the probability of this is:

(.4) * (.4) * (.4) * (.6) * (.6) = 0.02304

There are 9 other combinations of 3 'in-favors' and 2 'against' for at total of 10. You can discover these combinations on your hand or using an nCr calculator (online option - https://www.omnicalculator.com/statistics/combination)

So we can multiple this probability by 10 to get 0.2304.

Now, we can consider the case where 4 people are in favor. Let's consider the case where the first 4 people are in favor and the last one is against.

(.4) * (.4) * (.4) * (.4) * (.6) = 0.01536

There are 4 other combinations of 4 'in-favors' and 1 'against' for a total of 5 (can check on your hand or using nCr calculator). We can multiply this probability or percentage by 5 to get 0.0768.

Now there is the final possibility of all 5 being in favor, which there is only 1 possible combination of:

(.4) * (.4) * (.4) * (.4) * (.4) = 0.01024.

Now we can some all the probabilities in bold to get 0.31744 or 31.7%.