Scratch game odds

Dear Wizard of Odds,
I don’t gamble much at all but when I do, I like to have the best odds possible. My state lottery has a scratch game called Take Ten. Cost is $10 and there are 39 top prizes of $10,000 each remaining. And the probability of winning seems quite good (for gambling!): 1 in 9,696.97 of winning $10,000. The probability of winning the next highest prize of $100 is 1 in 2,307.69. (Of course, it’s also possible that the state printed out the wrong probabilities on its website.) But, assuming these probabilities are correct, how would I calculate the probability of winning EITHER $10,000 or $100? Do I add up the probabilities? So, is the probability of winning either of those two prizes 2 in 12,004.66 or 1 in 6,002.33??
Could you help me with a formula for calculating this? I think it’s probably pretty basic, but I just don’t know how to do it. Thanks.

Convert each to a probability and then add them up.

odds = 1/(1/9696.97 + 1/2307.69) = 1 in 1864

Quote: Jubilado

Dear Wizard of Odds,
I don’t gamble much at all but when I do, I like to have the best odds possible. My state lottery has a scratch game called Take Ten. Cost is $10 and there are 39 top prizes of $10,000 each remaining. And the probability of winning seems quite good (for gambling!): 1 in 9,696.97 of winning $10,000. The probability of winning the next highest prize of $100 is 1 in 2,307.69. (Of course, it’s also possible that the state printed out the wrong probabilities on its website.) But, assuming these probabilities are correct, how would I calculate the probability of winning EITHER $10,000 or $100? Do I add up the probabilities? So, is the probability of winning either of those two prizes 2 in 12,004.66 or 1 in 6,002.33??
Could you help me with a formula for calculating this? I think it’s probably pretty basic, but I just don’t know how to do it. Thanks.

The chance of winning the big prize is .00010312. Multiply that by $10,000 and you get $1.03125
The chance of winning $100 is .00043333. Multiply that by $100 and you get $.04333.
Add the two up and you get $1.07458.
So for every $10 you bet you would expect to get a little over $1 back via these two prize possibilities. There must be other smaller prizes you are not mentioning.

To answer your question, you add up the probabilities. .00010312 + .00043333 = .00053645, or around 1 in 1,864.

Thanks to both of you for your help! I actually gave you a wrong number in my question. The second prize is actually $500, not $100. The other prizes are $200 (1 in 515.02), $100 (1 in 220.59), $50 (1 in 60), $30 (1 in 30), $20 (1 in 15), and $10 (1 in 5.45).

I took the probabilities of winning a prize, by the amount of the prize, like you said, to get the expected return on $10. I did that for all the prizes and then added them up. I came up with $7.09 as what I could expect to get back for every $10 I spent, on average, considering the expected return for each prize. The overall probability appears to be .3072 and the odds seem to be 1 in 2.256. Does that sound about right?

For someone who has never calculated this before, it’s really an eye-opener. Think I’ll try to locate some other games and try some other calculations. THX!

Little trick with statistics... Most of the time when you're using words such as "either" or "or" then the result will be additive (i.e. this OR that). When you're using words like "both" or "and" then the results will be multiplicative due to their dependencies on each other (i.e. this AND that need to happen).

Most state lotteries have a return in the neighborhood of 50%. I'm guessing there is a typo on the website.

A lot of state post the odds on the back
Pa lottery usually say like 1 in 3.33 chance of winnin g

Quote: rsactuary

Most state lotteries have a return in the neighborhood of 50%. I'm guessing there is a typo on the website.

He says "39 prizes of 10k remaining". Different from the game at the start. Florida, for example, has a scratch off hotline, where you can check on how many top prizes remaining as of that moment for any given scratch off game before you buy cards. So if a game gets lopsided, you might go more heavily on it. He might have found one that is significantly above average, though 70% return is nothing to shout about either.