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The phrase "approximately the same chances" is correct and usable in written English.
It can be used when discussing the likelihood of two or more outcomes being similar in probability.
Example: "In this experiment, both methods have approximately the same chances of success, making either option viable."
Alternatives: "roughly equal probabilities" or "about the same likelihood".
Exact(2)
"I observed that I had approximately the same chances of achieving that position as of becoming an astronaut," he said.
In the social sphere, social positions, equally gifted and motivated citizens must have approximately the same chances at offices and positions, independent of their economic or social class and native endowments.
Similar(58)
Women deserve the same chances.
Everyone should be given the same chance'.
Give Harper the same chance.
Palestinians need the same chance.
Russia's persecuted gays deserve the same chance.
"Everybody there had the same chance," he said.
Almost anyone had the same chance of beating the market.
Not many young people have the same chance.
I have the same chance in China," Ma says.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com