How to determine which set of probabilities is a better predictor of the outcome? Suppose there are two columns of the probabilities of a certain outcome and then a column at the end indicating whether the outcome took place or not. Looking like this:
 prob_1    prob_2       Outcome
  0.49     0.54           1
  0.49     0.53           0
  0.48     0.48           1
  0.37     0.39           0
  0.54     0.53           1
  0.53     0.54           1

what would be the best way to determine which column is a better predictor of the outcome? I considered converting the raw probabilities into binary values (e.g if probability is greater than 0.5, I'd convert that to 1 and to 0 if the probability is less than 0.5) and then matching it with the outcome but that doesn't do great with for example 0.49 or 0.51 which are basically coin tosses being transformed into a sure thing one way or the other.
Any advice is appreciated!

Welcome to Cross Validated!
This is a common task in so-called classification problems in machine learning and statistical modeling. While plenty of people will be content to runs the probabilities to the nearest integer (so either zero or one), that puts you in a problematic situation where $0.51$ and $0.49$ are very different, but $0.51$ and $0.91$ are the same.
There are two common strictly proper scoring rules that measure the difference between claimed event probability values and the event occurrences: log loss and Brier score. These should work for your task, even if the probabilities are not generated from a machine learning model. In both of these cases, the lower the number, the better, and consider the logarithm to be the natural logarithm.
$$
\text{Log Loss}\\
-\sum_i \bigg[y_i\log(p_i)+(1-y_i)\log(1-p_i)\bigg]\\
\text{Brier Score}\\
\sum_i \bigg(y_i-p_i\bigg)^2
$$
