A Lottery Game With Conditional Probability?

[Calculuser]

Question: "In a lottery game each player tries to guess right 6 numbers designated in advance by choosing randomly from among numbers from 1 to 20. Given that one player guessed right 5 numbers out of 6 that he/she picked, what is the probability of guessing right the 6 numbers?"

The problem sounds to me an application of Bayes' Theorem, for we know "...one player guessed it right 5 numbers..." However, I couldn't figure out the cardinality of sets to be used in calculation. For instance, when we think of $S(5)$ as the set of 5 numbers guessed right and $S(6)$ as the set of 6 numbers guessed right, what are the cardinalities of $S(5)$ and $S(6)$ ($|S(5)|$ and $|S(6)|$)?

 

[phinds]

I often find I'm wrong on these things but this one seems to me to be simply that the fact that the other 5 of his choices are right is no longer a probability it is a certainty and thus the starting point is that you need one number, 1-20, to be right out of a possible set of choices of 1-20, which means the probability is 1/20.

 

[.Scott]

This is simpler than you think. In fact, it's going to be hard to help you without giving away the answer too readily.
One thing that is not explicitly stated is that the numbers in such a lottery are not reusable, so the winning list of 6 numbers will always contain 6 unique numbers - no duplicates.

I would imagine the person selecting his numbers in the same way that the winning number list was selected. He draws them from a jar one at a time. After 5 draws, he hasn't lost yet. What are his chances of getting the last one right?

(the answer is not 1/20)

 

[StoneTemplePython]

As is often the case, it is an underspecified / linguistic problem -- it depends in particular how we come to know that 5 numbers matched. So likelihood function needs pinned down.

It also isn't clear whether repeated numbers are allowed in the configuration.

 

[phinds]

It also isn't clear whether repeated numbers are allowed in the configuration.

Excellent point. My answer did not take that into account (or tacitly assumed duplication)

 

[Calculuser]

After re-reading the question in response to your replies above, I too noted its vagueness as some of you pointed out. The game is played as follows: six numbers, e.g. 1, 6, 18, 19, 8, 5, out of 1, 2, . . . 20 are first recorded, say on a piece of paper; and then then one player goes on to randomly pick six numbers, e.g. 11, 7, 12, 9, 20, 4, out of 1, 2, . . . 20, say the numbers have been imprinted on balls and are in a bag. With that, there is NO duplication of numbers.

 

[pbuk]

That clarifies one open question, but it doesn't help with the meaning of one player guessed right 5 numbers out of 6 that he/she picked.

This could mean, for instance:

• Exactly 5 of the 6 numbers picked matched winning numbers
• The first 5 numbers that were picked matched winning numbers
• The first 5 winning numbers matched numbers that were picked
• The first 5 numbers that were picked matched the first 5 winning numbers
• ...

People who enjoy solving interesting problems in probability often find this kind of under-specified linguistic problem frustrating.