Probability of the 1st number of a QP being 1 is:
1/39*5
The reason is if you pick a random number between 1 and 39, the chance of picking 1 is 1 in 39. Since 5 numbers are chosen and then printed in ascending order, where 1 would always be first since it's the smallest, there are 5 choose 1 (which is 5) different ways to do that.
Another way to look at this is:
prob(choose 1 then choose 4 more numbers) = 1/39 since after picking for the 1st number, the other 4 numbers must be larger than 1
prob(choose not 1, then choose 1, then choose 3 more numbers) = 38/39*1/38 = 1/39
prob(choose not 1, choose another not 1, then choose 1, then choose 2 more numbers) = 38/39*37/38*1/37 = 1/39
prob(choose not 1, choose another not 1, choose another not 1, choose 1, then choose 1 more number) = 38/39*37/38*36/37*1/36 = 1/39
prob(choose not 1, choose another not 1, choose another not 1, choose another not 1, then choose 1) = 38/39*37/38*36/37*35/36*1/35 = 1/39
then add them all together, and you get the same result, 1/39*5=5/39=0.128205
Now let's do a QP that has a smallest number of 2:
We need to choose 2 and choose 4 more numbers that are not 1 (since 2 must be the smallest number of the 5). That would be 1/39*37/38*36/37*35/36*34/35*5
1/39 is the probability of choosing 2. 37/38 is the probability of choosing a number from the remaining 38 that is not 1. 36/37 is the probability of choose a number from the remaining 37 that is not 1, and so on. At the end we multiply by 5 for the same reason as calculating the probability of a QP starting with 1, that is, calculate the probability that the first number chosen is the one you want, then multiply by 5 choose 1 (which is 5) because that particular number you want doesn't have to be the first one, it can be any one of the five.
Next we will do a QP that has a smallest number of 3:
We need to choose 3 and choose 4 more numbers that are not 1 or 2. That would be 1/39*36/38*35/37*34/36*33/35*5
1/39 is the probability of choosing 3. 36/38 is the probability of choosing a number from from the remaining 38 that is not 1 or 2. And so on.
Next we will do a QP that has a smallest number of 4:
We need to choose 4 and then choose 4 more numbers that are not 1, 2 or 3. That would be 1/39*35/38*34/37*33/36*32/35*5. Hopefully you can see the pattern.
The last one is a QP that has a smallest number of 5:
1/39*34/38*33/37*32/36*31/35*5
For QP that starts with 1, the probability 1/39*5 is 0.128205.
For QP that starts with 2, the probability using the arithmetic above is 0.11471
For QP that starts with 3, the probability is 0.102309
For QP that starts with 4, the probability is 0.090941
For QP that starts with 5, the probability is 0.080548
Clearly the probabilities are going down as we insist the lowest number be going up. At the end of this sequence would be an extremely low probability that the smallest of the 5 is 35, because if that is the case, then the numbers must be 35,36,37,38,39, which is the jackpot probability of 39 choose 5, or 1 in 575,757, which is 0.00000173684.
So, for a ticket with 5 lines where one starts with 1, another starts with 2, etc., (but in any order), the probability of that occurring is the product of the five probabilities above times 5 factorial. 5 factorial is 5*4*3*2*1 which is the number of ways you can order 5 different things.
0.128205*0.11471*0.102309*0.090941*0.080548*5*4*3*2*1 is 0.001323, or 0.1323%.
Take 1 divided by 0.001323 and you get about 756.
Therefore the probability of a 5 line QP starting with 1,2,3,4 and 5 (in any order) is 1 in 756.
Fairly unlikely but not astronomical by any means.