If the player has the SPADE:
1⁄3 of the time, he also has the HEART.
2⁄3 of the time he doesn’t, and so must choose the SPADE.
1⁄6 of the time he chooses the SPADE though he did have the HEART.
So 5⁄6 of the time he chooses the SPADE, but only 1⁄6 of the time does he choose the SPADE while having the HEART.
Thus, the chance of him having the HEART when he has chosen the SPADE is 1⁄5.
If the player has the SPADE:
1⁄3 of the time, he also has the HEART.
2⁄3 of the time he doesn’t, and so must choose the SPADE.
1⁄6 of the time he chooses the SPADE though he did have the HEART.
So 5⁄6 of the time he chooses the SPADE, but only 1⁄6 of the time does he choose the SPADE while having the HEART.
Thus, the chance of him having the HEART when he has chosen the SPADE is 1⁄5.