Determining the result of a BRV by successive probabalistic observations

caswell

[I have been struggling with this problem for weeks]

Suppose you are wondering about the outcome of a coin flip. You cannot observe the coin directly, but you have 3 tests you can perform to increase your confidence.

Before you perform any tests, all you know is p(heads) = 0.5.

Say the first test is to feel the surface of the coin. If it feels smooth, then there is a greater chance the coin is heads, suppose p(heads|smooth) = 0.7.

Say the second test is to run your finger nails across the surface. If you hear no scratching noise, then there is a greater chance the coin is heads, suppose p(heads|no scratchy noise) = 0.8.

Say the third test is to peak into the cup of your hands. If you see a slight glimmer in the darkness, then there is a greater chance the coin is heads, suppose p(heads|slight glimmer) = 0.6.

Also assume that each of these tests are independent.

What other assumptions can you make that allow you to estimated p(heads | (smooth) & (no scratchy noise) & (slight glimmer))?