Probability: Drawing two marbles simultaneously from the bag

You are correct. The problem with the teacher's method is that the various outcomes are not equi-probable.

Since the numbers are so small, we can write out all the possible, equiprobable, draw sequences (assuming we take all four in some order). They are: $$YYGG\quad YGYG\quad YGGY\quad GYYG\quad GYGY\quad GGYY$$

We see that exactly one way in six works.

Note that four out of six give mixed draws, demonstrating that the three cases your teacher points out are not equiprobable. Indeed, your method works for the mixed case as well: The first draw can be of either color and then the next gives a mixed draw with probabilty $\frac 23$.


Let's label the marbles for the sake of this question as 1 and 2 for each color $$Y_1, Y_2, G_1, G_2$$ If you choose two marble at the same time, the possible combinations are as follows:$$(Y_1, Y_2), (Y_1, G_1),(Y_1, G_2), (Y_2, G_1), (Y_2, G_2), (G_1, G_2)$$

This means there are $6$ unique outcomes, so the probability of the outcome we are looking for $(Y_1,Y_2)=\frac{1}{6}$. In this case you are correct as your teacher is lumping the middle 4 outcomes together as one "type" of outcome which is incorrect.

I hope this helps!


Well, your teacher is definitely wrong and here's why. You consider 6 cases each of which has the same probability. You could write numbers 1 and 2 on yellow and same on the green marbles and count all of the outcomes as if all of the marbles are different and we take them out one by one. You would then get 12 possible cases (4 for the 1st marble and 3 for the 2nd one), two of which fit your request - both of them are yellow (Y1, Y2 and Y2, Y1) so the probability is 1/6. She, however, considers 3 cases 2 of which are equal (obviously the probability to take out 2 yellows is the same as 2 greens), but the other one (marbles are different) is more probable as it can be achieved by not 2 as each of the previous ones (Y1, Y2 or Y2, Y1) but 8 ways: Y1G1, Y1G2, Y2G1, Y2G2,... well, I guess you can finish this yourself. I don't know about your country but we in Russia have a quite famous among mathematicians anecdote on exactly this theme: "-What is the probability to go out to the street and meet a dinosaur? -one half - either you meet it or you don't". While there can be a certain number of cases you always have to check that all of them are equally probable

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