1. ser improbable / que / Diego / comenzar / trabajar / hoy 2 Carmen y yo / dudar/que/ tú / conocer / a/el presidente
3. tal vez / llover / manana
4. ser verdad / que / yo/jugar/blen / al tenis

1. Es improbable que Diego comience a trabajar hoy.

2. Carmen y yo dudamos que tú conozcas al presidente.

3. Tal vez llueva mañana.

4. Es verdad que yo juego bien al tenis.

Explanation:

We must identify the subject, number and gender in each sentence, in this way it is possible to conjugate the verb in a logical way and that it has concordance with what you want to express.

1. ser improbable / que / Diego / comenzar / trabajar / hoy

Answer 1: Es improbable que Diego comience a trabajar hoy. (Subjunctive)

Translation 1: Diego is unlikely to start work today.

2. Carmen y yo / dudar/ que/ tú / conocer / a/ el presidente  

Answer 2: Carmen y yo dudamos que tú conozcas al presidente. (Subjunctive)

Translation 2: Carmen and I doubt that you know the president.

3. tal vez / llover / manana  

Answer 3: Tal vez llueva mañana. (Subjunctive)

Translation 3: Maybe it will rain tomorrow.

4. ser verdad / que / yo/ jugar/ bien / al tenis  

Answer 4: Es verdad que yo juego bien al tenis. (Indicative)

Translation 4: It is true that I play tennis well.

Explanation:

In Spanish when talking about things that are subjective and/or possible, but not certain, you have to use the subjunctive mood.

In Spanish when talking about things that are objective and/or certain you have to use the indicative mood.