Hello, Mike!

If I understand your question, there is no unique answer.

In a game of chance where the player knows the expected value (say -0.37),

how do you work out what the odds should be in order to break even in long run?

Without knowing the rules of the game,

. . the expected value is insufficient information.

Game 1

You flip a coin.

. . If you get Heads, you win $0.63

. . If you get Tails, you lose $1.00

The expected value is -$0.37.

I'm not sure how to "work out the odds"

. . since the nature of a coin flip cannot be changed.

We can, however, change the payoffs (and bet "even money").

Game 2

You roll a die.

. . If you get a "1", you win $0.78.

. . Otherwise, you lose $0.60.

The expected value is -$0.37.

Game 3

You randomly select a card from a full deck.

. . If it is an Ace, you win $1.19.

. . Otherwise, you lose $0.50.

The expected value is -$0.37.

Get it?