ksdjdj
Joined: Oct 20, 2013
• Posts: 909
February 28th, 2019 at 1:33:05 PM permalink
Some lucky person, won the 40 million Australian Powerball with the following numbers:

7, 5, 13, 4, 2, 14, 11 (main barrel)

FYI: 14 was the "Powerball number" (but it is not needed for the questions below)

1. For the "main barrel", what is the probability that the highest number would be 14 (or less)?

2. For the "main barrel", what is the probability that the highest number would be exactly 14? (edit/update: 145 pm)

"...Powerball is drawn from 2 barrels. 7 numbers are drawn from the main barrel (numbered from 1 to 35) and 1 Powerball number is drawn from the Powerball barrel (numbered from 1 to 20)."

The article below got me interested in asking the above questions.

https://www.msn.com/en-au/news/australia/one-lucky-winner-claims-massive-dollar40million-powerball-prize-but-some-who-missed-out-are-offended-by-the-ridiculous-winning-numbers/ar-BBUcIHa?li=AAgfLCP&ocid=mailsignout

Also, I don't know if this was the best spot to put this thread or "QUESTIONS AND ANSWERS - (gambling or math)", so Admin please feel free to move it, if necessary.

thanks
Last edited by: ksdjdj on Feb 28, 2019
ChesterDog
Joined: Jul 26, 2010
• Posts: 938
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February 28th, 2019 at 2:47:28 PM permalink
Quote: ksdjdj

...1. For the "main barrel", what is the probability that the highest number would be 14 (or less)?

2. For the "main barrel", what is the probability that the highest number would be exactly 14? ...

For 1, I get about 1 in 1959.

And for 2, I get about 1 in 3919.

The probabilities aren't as small as I would have thought. That must be because only 35 numbers are in the "main barrel."
Ayecarumba
Joined: Nov 17, 2009
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February 28th, 2019 at 2:54:18 PM permalink
The main barrel contains 35 numbers, each with an equal chance of being selected. Why are people "offended"?
Simplicity is the ultimate sophistication - Leonardo da Vinci
Chuckleberry
Joined: Jul 23, 2016
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February 28th, 2019 at 2:58:58 PM permalink
Here's what I have...
This is essentially a keno question, with different inputs. There are two ways to figure this out:
The more intuitive way is to think about each selection separately. Your initial draw will have a 14/35 chance of being a ball marked 14 or under. The second draw would have a 13/34 chance, then 12/33, and so on.
The chance of drawing 14 or less for all 7 balls in the next drawing would be: 14/35 x 13/34 x ... x 8/29 = .00051037 or about 1 in 1959.

The second way to do this is with combinations. (14 C 7)/(35 C 7) = .00051037

Building on #1, we know that 1 in 1959 draws will have every ball marked 14 or under. If we were to look at every combination of 7 out of 14 selected, half the time the #14 ball would be selected. So 1 in 3918 draws #14 will be the highest numbered ball drawn.

Just trying to stay positive
ksdjdj
Joined: Oct 20, 2013
• Posts: 909
February 28th, 2019 at 3:30:08 PM permalink
Quote: Ayecarumba

The main barrel contains 35 numbers, each with an equal chance of being selected. Why are people "offended"?

I don't know why (supposedly the person was joking though).

If that really did "offend" some people, then I would hate to see how "offended" they are if the main numbers were 1, 2, 3, 4, 5, 6 , 7.
ksdjdj
Joined: Oct 20, 2013
• Posts: 909
February 28th, 2019 at 3:48:11 PM permalink
Thanks everyone

There have been about 46 draws since the "7/35 Main and 1/20 Powerball" format was introduced.
So, the chance of "1. ...highest number would be 14 (or less)" over 46 draws is:
about a 1/43 chance that it happens at least once, right?
ThatDonGuy
Joined: Jun 22, 2011
• Posts: 4629
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February 28th, 2019 at 4:20:04 PM permalink
Quote: ksdjdj

1. For the "main barrel", what is the probability that the highest number would be 14 (or less)?

2. For the "main barrel", what is the probability that the highest number would be exactly 14? (edit/update: 145 pm)

"...Powerball is drawn from 2 barrels. 7 numbers are drawn from the main barrel (numbered from 1 to 35) and 1 Powerball number is drawn from the Powerball barrel (numbered from 1 to 20)."

Look at the problems this way:

1. You have 35 balls; 14 (the ones numbered 1-14) are red, and the other 21 are white. You draw seven red balls; what is the probability that all of them are red?
There are (35)C(7) = 6,724,520 sets of 7 balls that can be drawn, and (14)C(7) = 3432 sets of seven red balls; the probability is 3432 / 6,724,520 = about 1 in 1959.36.

2. This time, one ball (#14) is blue, 13 are red, and 21 are white. What is the probability that you draw the blue ball and 6 of the red ones?
Again, there are 6,724,520 sets of 7 balls; there are (1)c(1) x (14)C(6) = 3003 sets of six red and one blue balls, so the probability is 3003 / 6,724,520 = about 1 in 2239.27.
Chuckleberry
Joined: Jul 23, 2016
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February 28th, 2019 at 4:51:25 PM permalink
Quote: ksdjdj

Thanks everyone

There have been about 46 draws since the "7/35 Main and 1/20 Powerball" format was introduced.
So, the chance of "1. ...highest number would be 14 (or less)" over 46 draws is:
about a 1/43 chance that it happens at least once, right?

Yes, but you might be missing the reason why. The odds of it not happening at all over 46 draws is (1958/1959)^46 = 97.7%

So the odds of it happening at least once is 1 - 97.7% = about 2.3% or 1 in 43.
Just trying to stay positive
ChesterDog
Joined: Jul 26, 2010
• Posts: 938
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February 28th, 2019 at 5:53:26 PM permalink
Quote: ThatDonGuy

...2. This time, one ball (#14) is blue, 13 are red, and 21 are white. What is the probability that you draw the blue ball and 6 of the red ones?
Again, there are 6,724,520 sets of 7 balls; there are (1)c(1) x (14)C(6) = 3003 sets of six red and one blue balls...

Thanks for your clear explanation. I see a typo, though--"(14)C(6)" should be "(13)C(6)" because there are 13 reds.