Introduction:
In this problem, two persons P and Q are playing a game of tossing an unbiased coin alternately. The game continues until one of them gets a head. The person who gets the head first wins the game. P starts the game.

Approach:
To find the probability of P winning the game, we can analyze the possible outcomes of the game.

Possible Outcomes:
1. P gets a head on the first toss:
- In this case, P wins the game with a probability of 1/2.

2. P gets a tail on the first toss:
- In this case, the game continues and it becomes Q's turn.
- Now, the situation is similar to the original game, but with the roles of P and Q reversed.
- So, the probability of P winning from this point onwards is the same as the probability of Q winning in the original game.
- Let's denote this probability as P(Q).
- P(Q) represents the probability of Q winning the game starting from their turn.
- P(Q) = 1 - P(P)
- P(Q) = 1 - 1/2 = 1/2

Recursive Relationship:
Now, let's consider the probability of P winning the game, denoted as P(P).
There are two cases to consider:

1. P gets a head on the second toss:
- In this case, P wins the game with a probability of 1/2 * 1/2 = 1/4.
- The probability of this case occurring is 1/2, as P needs to get a tail on the first toss and a head on the second toss.

2. P gets a tail on the second toss:
- In this case, the game continues and it becomes Q's turn.
- The probability of P winning from this point onwards is P(Q) = 1/2.
- The probability of this case occurring is also 1/2, as P needs to get a tail on the first toss and a tail on the second toss.

Probability of P winning:
The probability of P winning the game can be expressed as follows:
P(P) = 1/2 + (1/2) * P(Q)

Solving the Recursive Relationship:
To solve the recursive relationship, we can substitute the expression for P(Q) into the equation for P(P):
P(P) = 1/2 + (1/2) * (1 - P(P))
P(P) = 1/2 + 1/2 - (1/2) * P(P)
2 * P(P) = 1 + 1/2
2 * P(P) = 3/2
P(P) = 3/4

Conclusion:
The probability of P winning the game is 3/4, which is equivalent to 0.75.
Since the correct answer is between 0.66 and 0.67, we can round the probability to 0.67, which is the most accurate approximation within the given range.