Lottery Probability Question?

Question

Assume there is a 5-numbered lottery. Would the probability of winning the lottery be higher if you chose the numbers [1,2,3,4,5] or when you choose any 5 random numbers? You can back up your argument with mathematical calculations.

My intuition says that the 5 random numbers would be higher, whereas my reason says that they would be equal. Can someone explain this to me?

Order in the lottery does NOT matter, therefore it is not a permutation but a combination.

Answer 1

It's a common fallacy to think a random string of numbers has a higher probability to show up than numbers we're familiar with, like 1-2-3-4-5. Mathematically, each number has an equal chance in life.

Also, pretend there were aliens who thought the string of numbers 1-5-8-4-2 was somehow special. They'd all think it'd be more unlikely to get this string than the string 1-2-3-4-5. Luckily, mathematics doesn't care one bit about human preference, so every string is equal, no matter how amazing it seems to us.

BUT, if you were buying a lottery ticket in real life, you might want to choose the string 1-2-3-4-5, since no one else would pick it because of their fallacy :)

Answer 2

The disparity between your intuition and reason is because you perceive the set {1,2,3,4,5} as having "low entropy", or a "high level of order", while a random 5-number set has "higher entropy". This is a concept often dealt with in advanced Probability Theory, i.e., that in open systems there is always a tendency towards higher entropy. However, the lottery is a closed system, where each number has equal chance of being chosen. Therefore, as is often seen in the famous "Gambler's Fallacy" problem, EVERY set of 5 numbers has an EQUAL chance of winning each drawing, INDEPENDENT of that 5-set's distribution or entropy, and INDEPENDENT of past and future drawings. Hope this helps,

Steve

Answer 3

Go with Janet P's answer. A lot of people don't understand this idea about probability, and many more like them. Imagine instead of your example [1,2,3,4,5] I picked MY favourite numbers less than 61 (or whatever the rules of the lottery specified), [3,5,17,32,27] <<<

Answer 4

The numbers [1,2,3,4,5] are no more or less likely than any other random combination of five numbers. However the numbers [1,2,3,4,5] are more likely to result in duplicate winners if they are chosen than a random combination of five numbers. This is because some people choose their numbers rather than let the machine pick them. And people don't choose randomly.

Answer 5

It would be equal with each number having equal chance to win.

You feel that random numbers are more like a winner because the chance of having the sequence 1,2,3,4,5 exactly is less than the chance of having other numbers' combinations.

Answer 6

Wow, this is really odd, and more fun than this looks. Let's say that I have 5 dices, each one numbered 1 to 6. There's 6^5 possible outcomes, we know that, that's a given. However, if we're going to choose a set of numbers to bet on, say, {1,2,3,4,5}, then the odds of it coming up is the permutation of the numbers divided by 6^5. Well, the maximum possible number of permutations is where all the numbers are different! This is so cool! Thanks for your suggestion!

An easy way to understand this is to look at what happens if I chose {1,1,1,1,1}. The odds of that happening is exactly 1 out of 6^5----not very good. I improve my odds by picking mixed numbers, and maximum odds where all of them are different.

Answer 7

I would have to say:
They are equal probabilities because
1) 1,2,3,4,5 are jsut as likely to pop up as any other random series you choose (eg. 6, 5, 4, 3, 9). Unless there is some kind fo a bias...

Answer 8

The probability to get 12345 is the same as to get any other 5 random numbers.

P = 1/(10^5 - 10^4).

Answer 9

_ _ _ _ _

10 10 10 10 10 = 10^5

means 100 000 possibilities
1 2 3 4 5 is just 1/100 000
just as likely as 55555

Answer 10

No idea.