What’s the difference between these two solutions of this probability question

binomial distributioncombinationspermutationsprobability

The problem: 8 identical balls are placed at random in three bags. What is the probability that first bag will contain 3 balls?

The solution I came up with:
We can distribute 8 identical balls among 3 different bags in 45 ways.
Out of those 45 ways, in 6 ways, the first bag will contain exactly 3 balls.
So, the probability is $\frac{6}{45} = \frac{2}{15} = 0.13333$

But with binomial distribution, and, this is how most of the people (on Quora) solved this question, and, this approach also seems to be pretty intuitive,
the probability that a ball will be placed in the first bag is 13. The probability that exactly 3 of the 8 balls will end up in the first bag can be found by using the binomial distribution:

8C3 x (1/3)^3 x (1 - 1/3)^5 = 0.273

I just wanna know, why the answers from these two approaches are very different?

Best Answer

The problem says the balls are placed "at random", but it doesn't specify the random distribution or process by which those balls are placed. The answer definitely depends on that information. For example:

  • If someone lists all 45 ways that the balls could end up, and then chooses one of those 45 ways uniformly at random, then the answer is $2/15$ as you said.
  • If, independently, each ball is placed in one of the three bags uniformly at random, then the answer is $1792/6561$ as the Quora answer said.

These aren't the only two possibilities. If someone flips a coin, and puts all the balls in the first bag if the coin is heads and puts all the coins in the second bag if the coin is tails, that is still placing the balls "at random", but the answer to the question would be $0$.

We always need to specify a probability distribution. If we intend the balls to be placed in the bags uniformly and independently, we should say that (even though that would be my best guess as to the writer's intentions).

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