Difficulty in explaining sample space

Question

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?

Answer 1

The (correct) sample space depends on how the probability problem has been framed:

• The set of letters in “$MISSISSIPPI$” is indeed $S_1=\{M,I,S,P\}.$

  In a probability experiment with sample space $S_1,$ $P(M)=\frac1{11};P(I)=P(S)=\frac4{11};P(P)=\frac2{11}.$

• On the other hand, we might prefer a classical probability experiment where all outcomes are equally likely. Distinguishing among different occurrences of each letter, our sample space is now $S_2=\{M,I_1,S_1,S_2,I_2,S_3,S_4,I_3,P_1,P_2,I_4\}.$

  Here, the event $\{I_1,I_2,I_3,I_4\}$ of choosing letter $I$ has probability $\frac4{11},$ whereas the outcome $I_2$ has probability $\frac1{11}.$

Answer 2

I don't agree with your explanation that there is any inherent difference in the bag of balls and the letters of Mississippi. I do think that the difference is how you define the sample space, ie as a space with equally like probabilities, or a set of distinguishable outcomes that do not have eqully likely probabilities.

If a sample space assumes that all probabilities are equally likely (which many treatments of probabilities do), then you would have to use S(1) ={M,I1,S1,S2,I2,S3,S4,I3,P1,P2,I4}
as your student suggested.

Some treatments of probability do not assume that all probabilities are equally likely and they might allow you to call the set of letters, S(2)={M,I,S,P} the sample space. However I would call it a set of distinguishable outcomes that do not have equally likely probabilities.