Introduction:
Turing machines are hypothetical devices that can simulate any algorithmic computation. They consist of a tape, a head that can read and write symbols on the tape, and a state register that determines the behavior of the machine.

Existential and Universal States:
Existential and universal states are special types of states in Turing machines that have specific properties:

1. Existential States:
- Existential states are states that are true if there exists at least one path through the computation that reaches the state.
- In other words, if there is any possible computation that leads to an existential state, the state is considered true.
- Existential states are often used in alternating Turing machines to represent the existence of a winning strategy in a game.

2. Universal States:
- Universal states are states that are true for all possible paths through the computation.
- In other words, if every possible computation leads to a universal state, the state is considered true.
- Universal states are often used in alternating Turing machines to represent the universal validity of a property.

Analysis of Turing Machine Types:
Let's analyze each of the given Turing machine types to determine if they have existential and universal states:

a) Alternating Turing machine:
- Alternating Turing machines are a type of non-deterministic Turing machines that alternate between two types of states: universal and existential states.
- The universal states represent the universal validity of a property, and the existential states represent the existence of a winning strategy in a game.
- Therefore, alternating Turing machines have both existential and universal states.

b) Probabilistic Turing machine:
- Probabilistic Turing machines are Turing machines that incorporate probabilistic choices in their transitions.
- However, probabilistic Turing machines do not have explicit existential and universal states.
- The behavior of a probabilistic Turing machine is determined by the probabilities assigned to different transitions, rather than the presence of existential or universal states.

c) Read-only Turing machine:
- Read-only Turing machines are Turing machines that can only read the input tape and cannot modify it.
- Similar to probabilistic Turing machines, read-only Turing machines do not have explicit existential and universal states.
- The behavior of a read-only Turing machine is determined by the transitions based on the input tape, rather than the presence of existential or universal states.

d) None of the mentioned:
- This option is incorrect because alternating Turing machines do have existential and universal states, as explained above.

Conclusion:
In conclusion, the correct answer is option 'A' - Alternating Turing machine. Alternating Turing machines have both existential and universal states, which are used to represent the existence of winning strategies and the universal validity of properties in games. Probabilistic Turing machines and read-only Turing machines do not have explicit existential and universal states.