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R.S. Regular user CT one day I'll have 184 Posts |
Math Studs,
I bought a lottery ticket the other day, and I'll need your help to make sense of one part of it. To play, you pick 4 numbers between 1 and 39, and then you pick a "Lucky Ball" number which is also between 1 and 39. So far, so good. Now, here's what I don't get - if the "Lucky Ball" can be any number from 1 to 39, then common sense says you have a 1 in 39 chance of getting the "Lucky Ball." However, the official odds as given says that the odds of getting the "Lucky Ball" are 1 in 61! I don't get it. Below I have pasted the odds from the lottery webpage. The last line says that you get a $5 payout if you get the Lucky Ball by itself, and it gives odds of 1 in 61 on the "Lucky Ball." What am I missing? Thanks. PS - Also, I'm curious as to how they compute the "overall" odds of 1 in 14.4. Extra credit if you can figure that out. LUCKY-4-LIFE PRIZE PAYOUT TABLE Match Prize Odds 4 Numbers + Lucky Ball $2,000 a week for LIFE 1 1 in 3,207,789 4 Numbers $10,000 2 1 in 84,416 3 Numbers + Lucky Ball $500 1 in 22,913 3 Numbers $50 1 in 603 2 Numbers + Lucky Ball $70 1 in 899 2 Numbers $4 1 in 24 1 Number + Lucky Ball $10 1 in 123 Lucky Ball Number $5 1 in 61 *Overall odds are 1 in 14.4. Top prize values subject to "split-prize" liability, and may be lower than shown. See Official Game Rules. Purchasers must be 18 or older. Ron
"It is error only, and not truth, that shrinks from inquiry." Thomas Paine
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S2000magician Inner circle Yorba Linda, CA 3465 Posts |
I haven't worked out everything, but there's a useful clue in looking at the "odds" (properly, "chance") of 4 Numbers + Lucky Ball, given as 1 in 3,207,789:
3,207,789 = 82,251 * 39 = 39C4 * 39C1. Thus, we know that the four (regular) Numbers are all different, and the Lucky Ball Number can be any number; i.e., the Lucky Ball Number could be the same as any one of the four (regular) Numbers. The short answer to the original question is that the chance of matching the Lucky Ball Number is 1 in 61 because the phrase "matching the Lucky Ball Number" means, more specifically, matching the Lucky Ball Number and not matching any of the other four (regular) Numbers. The probability of matching the Lucky Ball Number is 1/39; the probability of not matching the four regular Numbers is 35C4 / 39C4 = 52,360/82,251. These events are independent, so the probability of both is (1/39)*(52,360/82,251) = 52,360/3,207,789 = 1 / 61.26411 . . . . Thus, the chance of (only) matching the Lucky Ball Number is (approximately) 1 in 61. |
TomasB Inner circle Sweden 1144 Posts |
I wonder if they have guessed a probability of the player choosing the Lucky Ball as one of his other four numbers. Or, have they calculated the probability 1/61 for only those that has chosen five different numbers?
In the first case, I think that they would assume a probability of 4/39 of the player selecting the Lucky Ball as one of his previous four choices, but in reality I think that probability is far off. But they have to assume _something_ to even be able to calculate it. /Tomas |
S2000magician Inner circle Yorba Linda, CA 3465 Posts |
Tomas: You got your post in before I finished editing mine. Take a look at my last calculation.
I'm still not sure about the overall chance of 1 in 14.4, but it's late and I have to teach a finance class in the morning, so I'm off to bed. I'll revisit it later. The most important question has yet to be answered: Ron: did you win anything? |
R.S. Regular user CT one day I'll have 184 Posts |
Quote:
On 2009-09-19 02:39, S2000magician wrote: AH!!! I think that explains it. Thanks, S2000! So, counter-intuitively, I have MORE of a chance (1 in 14.4) of matching the Lucky Ball AND another number than I do of just matching the Lucky Ball by itself. Yes? And extending that reasoning, I would have MORE of a chance of matching 2 regular numbers plus the Lucky Ball? And perhaps, my chances are greater of matching ALL the numbers than of just getting the Lucky Ball! OK, now I'm confusing myself. Quote: -------------------------------------------------------------------------------- On 2009-09-19 03:15, S2000magician wrote: The most important question has yet to be answered: Ron: did you win anything? -------------------------------------------------------------------------------- Nope. Didn't match a single number. Apparently, the State of CT Lottery officials know too much about odds and probabilities. Those darned know-it-alls! Thanks, Ron
"It is error only, and not truth, that shrinks from inquiry." Thomas Paine
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S2000magician Inner circle Yorba Linda, CA 3465 Posts |
Quote:
On 2009-09-19 10:00, R.S. wrote: No. The proper conclusion is that you have a better chance of matching something other than just the Lucky Ball than you have of matching the Lucky Ball and nothing else. Unless I'm completely fuzzy-brained, I calculate the probability of matching nothing as (34C4 / 39C4) * (38 / 39) = 0.6203. (34C4 / 39C4 is the probability of not matching any of the four regular Numbers, and 38 / 39 is the probability of not matching the Lucky Ball Number.) Thus, the probability of matching something is 1 - 0.6203 = 0.3797 = 1 / 2.633. So I'd put the overall chance of winning as 1 in 2.633, not as 1 in 14.4. I'd appreciate someone checking this. It's late again. |
TomasB Inner circle Sweden 1144 Posts |
I understand now. It wasn't clear to me that the four balls drawn first was replaced for the Lucky Ball to be drawn.
If they have calculated the probabilities correctly, you can just sum all of them up which gives about 1/14.4. The reason the calculation in the post above of doing the compliment of matching nothing doesn't work is that you actually don't win anything for matching only one ball and no Lucky Ball. 1 - 35/39*34/38*33/37*32/36*38/39 - 4*4/39*35/38*34/37*33/36*38/39 which is about 1/14.37 /Tomas |
R.S. Regular user CT one day I'll have 184 Posts |
Thanks, guys. The Lottery should just post these odds instead:
4 Numbers + Lucky Ball $2,000 a week for LIFE = You're Dreaming 4 Numbers $10,000 = Yeah, right. 3 Numbers + Lucky Ball $500 = Very Very Difficult 3 Numbers $50 = Very Difficult 2 Numbers + Lucky Ball $70 = Difficult 2 Numbers $4 = One of These Days 1 Number + Lucky Ball $10 = Keep Trying Lucky Ball Number $5 = Maybe Ron :)
"It is error only, and not truth, that shrinks from inquiry." Thomas Paine
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TomasB Inner circle Sweden 1144 Posts |
The game actually seems quite fair, especially for young people.
If you die 38 years from now, the game is perfectly fair. If you live longer, the edge is on your side. I didn't calculate it exactly, but used the probabilities they've given: A game costs 2 dollars and you lose them 1 - 1/14.4 of the time, so your expected winnings (if you live 38 more years) for playing a single game is: 5/61 + 10/123 + 4/24 + 70/899 + 50/603 + 500/22913 + 10000/84416 + 2000*52*38/3207789 - 2*(1 - 1/14.4) > 0 But I just realized that the time scope is so large that you have to take into account that 2000 dollars gets less worth every year that passes. This _has_ to be taken into consideration. Would make a good puzzle, actually. /Tomas |
S2000magician Inner circle Yorba Linda, CA 3465 Posts |
Quote:
On 2009-09-20 05:41, TomasB wrote: Thanks! I missed that one. I shouldn't work on these late at night. |
R.S. Regular user CT one day I'll have 184 Posts |
You guys are awesome - thanks!
Ron
"It is error only, and not truth, that shrinks from inquiry." Thomas Paine
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johnmharris New user 58 Posts |
This thread provides a great example to use in a math class. Thanks!
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