# How to explain the sample space of Monty Hall problem?

I am now pretending to be a newbie student. I write the following sample space for the Monty Hall problem (It is a famous brain teaser, I assume you know it).

$$S=\{ (C,G1),(C,G2), (G1,G2), (G2,G1) \}$$

where the first tuple represents the first choice taken by the guest and the second tuple represents the optional switch offered by the host.

As a teacher I have difficulty to explain that $$\{(C,G1),(C,G2)\}$$ must be simplified to just one $$\{(C,G)\}$$ and $$n(S)=3$$. $$C$$ is the car, $$G1$$ and $$G2$$ are the goats.

Do you have any idea to explain it?

# Edit

• for $$(C,G1)$$, if you switch you will lose
• for $$(C,G2)$$, if you switch you will lose
• for $$(G1,G2)$$, if you switch you will win
• for $$(G2,G1)$$, if you switch you will win
I think the key here is that $$(C, G_1)$$ and $$(C, G_2)$$ are each only half as likely as each of the other two cases - and the standard "counting" approach to probability only works if all the cases are equally likely. To fix it, you need a third event - think of it as Monty flipping a coin. If it comes up heads, he picks the leftmost goat that's still hidden; if tails, the rightmost. If the player picks a goat, the coin flip doesn't matter, but it happens anyway. This gives us six outcomes: $$(C, H, G_1), (C, T, G_2), (G_1, H, G_2), (G_1, T, G_2), (G_2, H, G_1), (G_2, T, G_1)$$.