When one looks at the default [listed first by the non-historic dictionaries Cambridge, Collins, Britannica, OLD, though not AHD and Wiktionary] sense of the adjective, contingent:

contingent [adjective]:

[1] dependent on or conditioned by something else

• Payment is contingent on fulfillment of certain conditions.
• a plan contingent on the weather

[2] likely but not certain to happen; [quite] possible

...

[Merriam-Webster]

one sees that there is at least a reasonable pull towards the resultative (of conditions obtaining, often though not always as a result of human activity, often undesirable) sense (admittedly not the default sense) for the noun:

contingency [noun]:

1: a contingent event or condition, such as

a: an event (such as an emergency) that may but is not certain to occur

• trying to provide for every contingency

b: something liable to happen as an adjunct to or result of something else

• the contingencies of war

[+ other senses]  [Merriam-Webster]

The synonym 'possibility' does not have this ... denotation? connotation?

When one looks at the default [listed first by the non-historic dictionaries Cambridge, Collins, Britannica, OLD, though not AHD and Wiktionary] sense of the adjective, contingent:

contingent [adjective]:

[1] dependent on or conditioned by something else

• Payment is contingent on fulfillment of certain conditions.
• a plan contingent on the weather

[2] likely but not certain to happen; [quite] possible

...

                                                                                                    [Merriam-Webster]

often, but not always, followed by an on-phrase:

• Social change and historical events are highly contingent processes, in a specific sense: they are the result of multiple causal influences that "could have been otherwise" and that have conjoined at a particular point in time in bringing about an event of interest.                                      [Daniel Little; Understanding Society]

one sees that there is at least a reasonable pull towards the resultative (of conditions obtaining, often though not always as a result of human activity, often undesirable) sense (admittedly not the default sense) for the noun:

contingency [noun]:

1: a contingent event or condition, such as

a: an event (such as an emergency) that may but is not certain to occur

• trying to provide for every contingency

b: something liable to happen as an adjunct to or result of something else

• the contingencies of war

                [+ other senses]                                                [Merriam-Webster]       [bolding mine]

The synonym 'possibility' does not have this ... denotation? connotation?

When one looks at the default sense of the adjective, contingent:

contingent [adjective]:

[1] dependent on or conditioned by something else

• Payment is contingent on fulfillment of certain conditions.
• a plan contingent on the weather

[2] likely but not certain to happen; [quite] possible

...

[Merriam-Webster]

one sees that there is at least a reasonable pull towards the resultative (of conditions obtaining, often though not always as a result of human activity, often undesirable) sense (admittedly not the default sense):

contingency [noun]:

1: a contingent event or condition, such as

a: an event (such as an emergency) that may but is not certain to occur

• trying to provide for every contingency

b: something liable to happen as an adjunct to or result of something else

• the contingencies of war

[Merriam-Webster]

When one looks at the default [listed first by the non-historic dictionaries Cambridge, Collins, Britannica, OLD, though not AHD and Wiktionary] sense of the adjective, contingent:

contingent [adjective]:

[1] dependent on or conditioned by something else

• Payment is contingent on fulfillment of certain conditions.
• a plan contingent on the weather

[2] likely but not certain to happen; [quite] possible

...

[Merriam-Webster]

one sees that there is at least a reasonable pull towards the resultative (of conditions obtaining, often though not always as a result of human activity, often undesirable) sense (admittedly not the default sense) for the noun:

contingency [noun]:

1: a contingent event or condition, such as

a: an event (such as an emergency) that may but is not certain to occur

• trying to provide for every contingency

b: something liable to happen as an adjunct to or result of something else

• the contingencies of war

[+ other senses] [Merriam-Webster]

The synonym 'possibility' does not have this ... denotation? connotation?