Step-by-step explanation:
based on my understanding of the dreidel, this is a regular spinner with 4 equal sides (but different symbols on them, so, we can clearly distinguish between the possible 4 different outcomes).
in other words, this resembles a die with only 4 sides and therefore only 4 equally probable outcomes.
and therefore, the probability to get the specified symbol in 1 attempt is indeed 1/4.
remember, a probability is always desired cases over totally possible cases.
and here, we have one desired outcome out of 4 possible outcomes. hence the probability of 1/4.
now, we are spinning the dreidel twice.
that means we have now 4×4 = 16 possible outcomes.
but we only want the outcomes, where the specified symbol is showing exactly once - either after the first spin or after the second spin.
so, out of the 16 possible combinations, we want only the ones, where the first spin delivered that result :
1 option on the first spin and 3 options (4 minus the already delivered result) on the second spin : 1×3 = 3.
and the ones, where the second spin delivered the result (but not the first). so, we have 3×1 = 3 options there.
that means we have 6 desired cases out of total 16 possible outcomes, and the probability is
6/16 = 3/8
mathematically we would simply say that the probability is the sum of
the probability of spinning it first combined with the probability of not spinning it second.
the probability of not spinning it first combined with the probability to spin it second.
that is
1/4 × 3/4 = 3/16
+
3/4 × 1/4 = 3/16
-----------------------
6/16 = 3/8
I don't know the options you can select, but I hope you understand the principles I explained. and I gave you the result and the way to calculate it.
so, hopefully you find the fitting options.
can be parameterized by
and 
. Then the surface integral can be computed with
