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So let's start with the real question I am trying to figure out.

There is a youtube channel that I frequent, where a Let's Player picks a random game to play by spinning a wheel. I don't recall how many spaces there are off-hand, but let's say there are 20 of them. Three out of these 20 spaces are labeled "Request" or "Free Pick", meaning he will play a game off of the request list from people who have requested a specific game, or pick one himself. So obviously there's a 3:20 chance of him picking something off that list or picking a game himself.

Now, this wheel is accompanied by a special mascot character, roughly 50 different ones in total that appear randomly.

Recently, he has added a new mascot that has nothing but Request and Free Pick, meaning there is a 1:50 chance, every time he chooses to spin the wheel, that he will do a request or free pick.

Now, ignoring the possibility of choosing a request at random, how much has he increased his odds of doing a request/free pick by adding this new mascot?

Or, to put it simply, if you have a 3:20 chance to win, but you then add a 1:50 chance to win prior to your 3:20 chance, by how much have your odds increased? How much were your initial odds, and what are they now?

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As you say, your initial odds were $\frac 3{20}=0.15$. Now you have $\frac 1{50}=0.02$ to win the mascot, and if you lose (probability $0.98$) you have the same $0.15$. So the total chance is $0.02+0.98\cdot0.15=0.167$ You would add them, but have double counted the times you win both.

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This is the exact answer I was looking for. Thank you. :) I knew it couldn't just be as simple as adding the two probabilities together, but I couldn't for the life of me figure out what the proper way to combine the odds was. This makes perfect sense. Which means the increase in odds is 0.012, or a 1.2% increase (in percentage points, not as a percent of the original odds). – Zibbobz Aug 22 '13 at 14:12
Oops! That should be 0.017, not 0.012. So a 1.7% (Again, percentage points) increase, not 1.2%. – Zibbobz Aug 22 '13 at 14:20

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