I think that OP is confusing expected value with probability.
The expected value formula is the probability of an event multiplied by the amount of times the event happens: (P(x) * n).
This explains the P(B) = 1.5 the OP put above—he means the expectation is 1.5, because P(waking with any one coin flip result) = 1⁄2 and the times it occurs is 3.
So the halfers believe the expectation is 1⁄2 for waking with heads and 1⁄2 for waking with tails.
The thirders have the expected values right: 1 for waking with heads, 2 for waking with tails.
I think that OP is confusing expected value with probability.
The expected value formula is the probability of an event multiplied by the amount of times the event happens: (P(x) * n).
This explains the P(B) = 1.5 the OP put above—he means the expectation is 1.5, because P(waking with any one coin flip result) = 1⁄2 and the times it occurs is 3.
So the halfers believe the expectation is 1⁄2 for waking with heads and 1⁄2 for waking with tails.
The thirders have the expected values right: 1 for waking with heads, 2 for waking with tails.