To find the total probability, we add up the probabilities of each of the events.
For example, the total probability of getting the jackpot in 4 lines in a 6/48 game would be 1/12271512 + 1/12271512 + 1/12271512 + 1/12271512
=4/12271512.
The probability of getting the jackpot in 8 lines would be 8/12,271,512.
This is the probability for 8 random lines, I think we can agree on that. Now the question becomes what is the probability given your strategy?
First, we look at the two lines with no winning numbers. Naturally, the probability is 0.
Next, let's take a line from the 6 that are left:
The first number has a 6/36 chance of being picked. Then 5/35, because one number is removed from the set.
The probability of this line then becomes 6/36*5/35*4/34*3/33*2/32*1/31 = 1,947,792.
This is where the confusion comes from. It looks like that the probability of the lines from those 6 have a probability of 1/1,947,792, so your
total probability would be 6/1,947,792, which is much higher than 8 random lines, right?
Wrong.
The reason is, that's only the probability of one of the lines from the 6 lines with the winning numbers, not ALL of them.
You look at it this way.
The probability of a winning number appearing in the first line is
1 - 30/36*29/35*28/34*27/33*26/32*25/31 = 1 - 1/2*29/7*7/17*3/11*13/8*25/31 = 1 - 197925/649264 = 451339/649264
This calculation shows the total probability minus the probability of none of the 6 winning numbers appearing. It shows that at least 1 winning number
will be in line 1 roughly 70% of the time.
In this case, none of the other 5 lines will be a jackpot winner, because at least 1 winning number is not in that line.
In the 197925/649264 times where the first line does not have a winning number, the probability for line two to have a winning number is
1 - 24/30*23/29*22/28*21/27*20/26*19/25 = 1/5*23/29*11/1*1/9*1/13*19/5 = 1 - 4807/84825 = 80018/84825 which is approximately 94.3%.
So in the case that the first line had no winning numbers, 94.3% of the time there will at least be one winning number in line 2.
This means that approximately 98.7% of the time, lines 3-6 have a 0 probability of winning!
You might skip through all that math, because it gets confusing very quickly. That's why I'll continue the rest in another post, for anyone
who cares. The point here is, you're not making every line in the 6 possible lines better. As you saw from above, 3-4 of the lines have 0%
chance to win the vast majority of the time. So when you balance it all out, it has the same probability as 8 random lines.