A bag contains 10 identically-shaped marbles: 3 red, 4 blue, and 3 green. A person randomly pulls out the marbles one by one (without replacement).
I'm not understanding why the probability of picking a blue marble 3rd is 4/10 – I'm trying to calculate the probability, and just not getting 4/10 though I understand that's the correct answer. Can someone please assist? Thank you!
In the calculation above, I'm trying:
- if all three are blue;
- if the first is not blue but the next 2 are blue;
- if the first two are not blue but the 3rd is blue;
- if the first is blue, second is not blue, and third is blue
As a note, I aggregated the red + green marbles as I didn't think their colors mattered specifically, just that 6 marbles are not blue
Best Answer
To summarize the discussion in the comments:
The methodology is solid, but there is a simple arithmetic error. Specifically, the cases $\#2$ and $\#4$ should give the same value. Both should be $\frac {6\times 4\times 3}{10\times 9\times 8}$ with some permutation of the factors in the numerator.
Of course, it's better to simply remark that each ball has an equal chance of being chosen in each position, thus each ball has a $\frac 1{10}$ of being chosen fourth, and we just multiply by $4$ because there are four blue balls. Of course, this applies equally well to each position, there is nothing special about the fourth position.