Probability Puzzle With Clever Solution: Increasing Lottery Numbers

A lottery draws 5 different numbers from the numbers 1 to 60. What is the probability the numbers are all increasing or all decreasing? What if a lottery ticket draws k distinct numbers from 1 to n? Watch the video for a solution.

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[mry]

Don't they always put them in increasing order on the ticket?

[CyclopGamingChannel]

Would be long to explain but if i'm correct i'll do so after finding it out, my conclusion is that n doesn't matter, and the probability is 2/k!, thus for the first problem the solution would be 2/5!=2/120=1/60=0.01(6)

Edit 1: Alright so after watching the whole video it seems i was correct.. but the way i came to my conclusion took way longer than in the video. The way i did it was a somewhat brute start: I wrote down all possible drawing cases, for some small values for n and k. Soon after writing two of the possible lotteries, i've started noticing some interesting patterns, such that the probability of ascending matches the probability of descending number drawn. Also that for any n, where k=1 or k=2, the probability is 1 [100%]. And soon after, even though the rule says that k<n, i still tried out a couple possibilities where k=n and quickly after arrived that for any n where k=n, the probability is 2/k!, it was simple to arrive at this, same as it was in the video. But at this point i wasn't aware of the fact that n didn't matter. And so i started writing bigger possible lotteries and their possible drawings. To eventually arrive at the conclusion that n doesn't matter i had to write down for n=4, k=3 and n=5, k=3 where i then noticed the pattern that the answer doesn't care about n, because n only increases the amount of different possible draws but not their chance of being ascending or descending. And thus going the long way i arrived that my solution for any n where k=n was the solution for it all.

[ShinySwalot]

1:32 You silly man, A, B, C, D & E aren't numbers, they are letters!

[Eric-rcvg]

Thank you so much, I have mathcounts coming up and I always have trouble with these problems. 😁😊

[benvanderzwan]

This one was super easy. Originally I thought I had to factor in the total possibilities for 60 numbers to be arranged with repeats but than realized it was to a distinct set of 5.

[TUH]

How would you solve this?

Let's say we have 1000 x 1000 grid and 1000 different (x, y) coordinates from that grid are chosen randomly. Then we choose 3 coordinates for the edges of the triangle (the most favourable ones ofc). What is the probability that area of that triangle is an integer?

[mulimotola]

What about the other 55 possible numbers of the total 60? Aren't they considered as separate outcomes, and thus need to be taken into account in the calculation? Drawing 1, 2, 3, 4, 5 is different than 2, 3, 4, 5, 6

[CyclopGamingChannel]

What happens for k=1? Can a single number be considered both ascending and descending? Is it neither? If prior then the probability is 1 and thus the proposed formula doesn't work because 2/k!=2/1!=2 and 2=/=1. If it's the latter then the probability is 0 and again 2=/=0. Or since we're looking for ascension and descension, does that imply that k cannot be 1?

[lucabastianello]

Here you haven't considered the first number that is what can change the amount. The first numebr has the same probability to come out, but then change drastically. Cause id you have from 1 to 60 and you draw 1 or 60 the probability is completely different from drawing a number in the middle.

[stuartyellow]

Did i just understood the question totaly wrong or did u eplained it all wrong?
I dont understand ur explination for k. Why there are only 2 successfull? Frst of all for k=1 there is only 1 success and the probability is 1 and not 2.
And if k<n u cant use ur example and there are more than 2 successes possible every time!
(my english is bad. Im sorry for that)

[atharvas]

What if not all k numbers are drawn?
Say a bag has balls numbered 1 to 100. What is the probability that any, say 5, randomly drawn balls will be in a strictly increasing or decreasing order.
Gererally:
A bag has balls numbered 1 to n. What is the P(any k drawn balls-with(out) replacement-will be in a strictly inc or dec order) ?

[david]

I framed the problem in my head as:

"Given 60 numbers and 5 remaining picks, what are the odds of picking a number N1 that will still allow you to pick 4 more numbers that are greater than N1 (i.e. 56 or less)",

then "Given 60-N1 numbers, and 4 remaining picks, what are the odds of picking a number N2 that will still allow you to pick 3 more numbers that are greater than N2?"

Then, "Given X numbers, and Y remaining picks, what are the odds of picking a number N that will still allow you to pick (Y-1) more numbers that are greater than N?"

Then I figured that if I could get a handle on that formulation, I could solve this problem recursively.

Then I gave up.

Then I felt like an idiot after watching this video.

[lifeaschrisyanak]

This problem was too easy for me.
🇬🇧👍🏻🇬🇧

[jilow]

If two numbers could be identical then there could be more than 1 way to put them in increasing order. still be 120 ways to arrange the 5 objects. For example suppose all 5 numerbers were 9. then either there is 0 ways to put them from increasing to decreasing order, or we count all 120 possible ordering and say there is 100% chance.

[ellinorjonson]

Aren't there several ways to order a set if two or more of the numbers are equal?

[necrolord]

The number of ways to draw the numbers all in order, is 1 per combination of numbers. Therefore, the number of ways to draw them in order is nCk.

The number of ways to draw the numbers overall is nPk.

Therefore, the probability of the numbers all being in order is nCk / nPk = (n!/(n-k)!k!)/(n!/(n-k)!) = 1/k!

In the problem presented, that would be 1/5! = 1/120 which is approximately

[dannyspeagle]

Presh Telwalker 

I actually thought the question, as asked, was "what is the frequency in which lotteries are drawn in increasing/decreasing order."  But then you answered that more general question by addressing--what I think is--a much simpler question, namely "what is the frequency in which the numbers 1 through 5 could be drawn."

I could be mistaken, but I strongly believe answering the first question, as asked, is FAR more complex than the answer you gave.

For example, I believe the answer to the question, as originally asked, has to take into account that the population from which numbers are being drawn from is, say ping pong balls numbered from 1 to 60.

Could you address this?

I would be fascinated to see your take on addressing the more complicated (and realistic) setup.

[alexanderhoward]

There are 60*59*58*57*56 ways of drawing numbers. That means there are 655381440 combinations. Then you have to find how many of them are of increasing order, multiple it by 2 then divide by this number.

[bryanmico]

the gen. formula is:
[(2^(n-r))(n)]÷[number of ways or permutation]
where n= larger given number
r= smaller

[kezzyhko]

So for 1 distinct number the probability will be 2/1! which is 200% :D