I am doing a data course on statistics and in one of the questions, they're suggesting that the statements given below are mathematically the same. I disagree. Can they be mathematically proven to be the same?
If we invest to improve safety in Midtown, there is a 25% chance we'll save $1,000,000 and a 75% chance we won't save any money.
If we invest to improve safety in Midtown, we are certain to save $250,000.
"If we invest to improve safety in Midtown, there is a 25% chance we'll save \$1,000,000 and a 75% chance we won't save any money."
$$\implies \mathbb{E}[saving] = 0.25(1,000,000) + 0.75 (0) = 250,000$$
"If we invest to improve safety in Midtown, we are certain to save $250,000."
$$\implies \mathbb{E}[saving] = 1 (250,000) = 250,000$$
As you can see, the statements produce a different weighted sum. But those weighted sums both evaluate to the same number.
That is probably what the question means by "mathematically equivalent". Not to say they worded the question well at all :)