I am running into problems explaining to my students in high school what exactly is sample space in probabilities, especially with identical objects.
For example, according to Q6.2.3 of this UIC website, the sample space for a letter is chosen at random from the word “MISSISSIPPI” is S={M, I, S, P}. The reason given was the letters, for example S, are indistinguishable.
However, I have also seen solution to question such as:
“Write down the sample space of the experiment of drawing a marble at random from a bag containing 5 identical blue (B) marbles and 4 identical red (R) marbles.”
would be S={B1,B2,B3,B4,B5,R1,R2,R3,R4} but not S={B,R} like in the example above.
When a student asked me why is the marbles in the second example distinct if they are identical and I couldn’t really give a convincing answer. I explained that in the first example, we are not drawing a letter from a bag but instead we are picking a letter from the word right in front of us. So we either choose a M, I, S or P. But in the second example, the marbles are in a bag so we can’t see the balls. And when we pick, we don’t know what we will be drawing. But when a student asked “then wouldn’t the second example be S={B,R} since we either pick up a blue or red marble.
A second student then asked then why can’t we distinguish the letters in the first problem such that S={M,I1,S1,S2,I2,S3,S4,I3,P1,P2,I4}
I was a bit dumbfounded and could not give a convincing explanation. Any suggestions how should I better explain it?