das Mittel
remedy
A substance or preparation used for a specific purpose, e.g. for healing, cleaning, or combating pests.
Gibt es ein wirksames Mittel gegen Mückenstiche?
Is there an effective remedy for mosquito bites?

remedy / means
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remedy
A substance or preparation used for a specific purpose, e.g. for healing, cleaning, or combating pests.
Gibt es ein wirksames Mittel gegen Mückenstiche?
Is there an effective remedy for mosquito bites?

'Das Mittel' (remedy/means) is closely related to 'die Mitte' (the middle). Figuratively speaking, a 'Mittel' is something that stands in the middle between a problem and a solution and helps you get from one to the other.
Dieses Medikament ist ein gutes Mittel gegen Kopfschmerzen.
This medication is a good remedy for headaches.
'Mittel' (remedy/agent) is often the second part of compound nouns. The first part describes the purpose, e.g. 'Schmerzmittel' (painkiller, against pain), 'Waschmittel' (detergent, for washing), or 'Lösungsmittel' (solvent, for dissolving).
Hast du ein gutes Waschmittel für bunte Wäsche?
Do you have a good detergent for colored laundry?
The idiom 'ein Mittel zum Zweck' (a means to an end) is very common. It describes an action or thing that is not good in itself, but is necessary in order to achieve a specific goal.
Für ihn war der Job nur ein Mittel zum Zweck, um Geld zu verdienen.
For him, the job was only a means to an end: to earn money.
The meaning 'Durchschnitt' (average) is used almost only in mathematical or scientific contexts. Watch for words like 'arithmetisch' (arithmetic) or the phrase 'im Mittel' (on average).
Im Mittel beträgt die Temperatur hier im Sommer 25 Grad.
On average, the temperature here in summer is 25 degrees.
possibility
'Möglichkeit' and 'Mittel' can both have to do with a way to reach a goal. 'Möglichkeit' means an option or opportunity: something you can do. 'Mittel' more often means the method or action with which you achieve a goal. So 'Möglichkeit' is more about the available options, while 'Mittel' is more about the how. Learners may confuse the words because the expression 'Mittel und Wege' (ways and means) also makes you think of possibilities. In clear sentences, however, the important distinction is: you can have a 'Möglichkeit', but you use or apply a 'Mittel'.

Geduld ist oft ein gutes Mittel, um einen Streit zu lösen.
Patience is often a good means of resolving an argument.
Zu Hause habe ich die Möglichkeit, in Ruhe für den Test zu lernen.
At home, I have the opportunity to study for the test in peace.