I'm looking at some probabilities, and I think I'm doing the math correct, but the results seem a little counter-intuitive. Perhaps someone can comment.

These examples are for a chance to score a hit on d6.

If you increase the chance to hit by one die face, it's a flat increase of 16.7 percentage points, no matter if increasing from 1/6 to 2/6, or from 4/6 to 5/6.

Easy enough.

But say you have a 4/6 chance to hit, and instead of increasing to 5/6, you add a second die with a 1/6 chance to score a hit as well, and roll both dice.

In this case the additional chance to hit is less than 16.7 points, meaning +1 die face on the original die is worth much more than adding a face on an additional die. And the added chance seems to vary depending on your original chance. (It's worth more if your original chance was 1/6 than if 5/6)

This seems a bit counter-intuitive to me, since at first blush both examples seem to add one scoring die face to the mix.

So, either my math is wrong, or probabilities are just weird that way.

Here's my math:

Example 1 (Add a scoring face on same die):

4/6 (66.7%) -> 5/6 (83.3%)

Increase of 16.6 points

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Example 2 (Add a scoring face on a new die):

4/6 (66.7%)

4/6 + 1/6 (72.3% of at least one hit)

Increase of 5.6 points

Example 2 breakdown of ways to hit with 2 dice:

hit on main and miss on secondary = 4/6 * 5/6 = 55.6%

miss on main and hit on secondary = 2/6 * 1/6 = 5.6%

hit on main and hit on secondary = 4/6 * 1/6 = 11.1%

Total = 72.3%

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But if the original began with 2/6 chance instead of 4/6, that additional die is now worth more, but still not as much as an increase to 3/6.

Example 3 (Add a scoring face on a new die):

2/6 (33.3%)

2/6 + 1/6 (44.4% of at least one hit)

Increase of 11.1 points

Example 3 breakdown of ways to hit with 2 dice:

hit on main and miss on secondary = 2/6 * 5/6 = 27.7%

miss on main and hit on secondary = 4/6 * 1/6 = 11.1%

hit on main and hit on secondary = 2/6 * 1/6 = 5.6%

Total = 44.4%

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And of course there's more disparity between the original starting with 1/6 compared to 5/6

Am I doing this correctly?

EDIT: Copied some numbers wrong from my notes.

Ok, so I'm doing the math right, but just did it the long way. Good to know.

I did, however, copy some numbers wrong from my notes which I fixed in the original post. Example 3 now shows a larger percentage point increase.