Help please

[Steph]

Hi my son (11) had a question about probability, which is beyond my mathematical capability. We were playing Rummikub and he had a therory that being the second person to pick up tiles would mean he has a greater chance of picking up a joker.

Rummikub Problem:
There are 106 tiles, 2 of those tiles are jokers and each player picks up 14 tiles. The player who picks up the highest tile at the start gets to go first and gets to pick up their 14 tiles before the other players. Is this actually an advantage, or would you be more likely to pick up a joker if you picked up 2nd, 3rd, 4th…?

* The player who picks up the highest number can also decide whether they want to keep that number or put it back.

Any help would be much appreciated!!

Thank you

 

[Dr.Peterson]

Hi my son (11) had a question about probability, which is beyond my mathematical capability. We were playing Rummikub and he had a therory that being the second person to pick up tiles would mean he has a greater chance of picking up a joker.

Rummikub Problem:
There are 106 tiles, 2 of those tiles are jokers and each player picks up 14 tiles. The player who picks up the highest tile at the start gets to go first and gets to pick up their 14 tiles before the other players. Is this actually an advantage, or would you be more likely to pick up a joker if you picked up 2nd, 3rd, 4th…?

* The player who picks up the highest number can also decide whether they want to keep that number or put it back.

Any help would be much appreciated!!

Thank you

What is the reason for thinking the second person has a better chance? Is he thinking about the fact that the first person may be keeping a tile that definitely is not a joker, or is he not taking that rule into account, and just thinking about choosing tiles?

In the latter case, you might want to imagine that each player has chosen tiles, in the appropriate order, without turning them over. At that point, is any person more likely to have a joker?

 

[blamocur]

According to Wikipedia players pick their 14 tiles completely randomly. I am betting there are no advantages to either player.

 

[Steph]

What is the reason for thinking the second person has a better chance? Is he thinking about the fact that the first person may be keeping a tile that definitely is not a joker, or is he not taking that rule into account, and just thinking about choosing tiles?

In the latter case, you might want to imagine that each player has chosen tiles, in the appropriate order, without turning them over. At that point, is any person more likely to have a joker?

What is the reason for thinking the second person has a better chance? Is he thinking about the fact that the first person may be keeping a tile that definitely is not a joker, or is he not taking that rule into account, and just thinking about choosing tiles?

In the latter case, you might want to imagine that each player has chosen tiles, in the appropriate order, without turning them over. At that point, is any person more likely to have a joker?

Hi Dr Peterson thank you for your response. So when players pick up their 14 tiles the other players do not see each other’s tiles. My sons thoughts were that if the first person picks up 14 tiles then they have a 2/106 chance of getting a joker and when the second person picks up 14 tiles they then have a 2/92 chance because he felt like it was unlikely that the first person would have picked up a joker.

 

[Steph]

According to Wikipedia players pick their 14 tiles completely randomly. I am betting there are no advantages to either player.

Yes so players pick up 14 tiles from a pool of 106 tiles and no other players can see each other’s tiles. If each player was to pick in order would the first player have a better chance of picking up a joker vs. the second player? My son was stuck on 2/106 for the first player, 2/92 for the second player and I didn’t know how to explain that the second players probability wouldn’t be 2/92 because you haven’t factored in the probability of the first player picking up either 1 or 2 jokers.

 

[Steven G]

This is basically what Dr Peterson said.
Suppose there are 4 players which means that they will pick up 4*14 = 56 tiles. Since each player has no idea of the value of the tiles they are picking the following setup would be the same.
Someone picks out 56 tiles and gives out 14 to each player. In the 4 groups of 14, since they were randomly chosen, each group has the same chance of getting 0 jokers, 1 joker or both jokers.

Why is your son wrong? Your son is assuming that the 1st player did not get any jokers. If this is correct, then there are 2 jokers in the remains 92 tiles. Another problem with your son's response is if there are 92 tiles including 2 jokers, then if he picks 1 tile (Not 14!) he would have a 2/92 chance of getting a joker.
Suppose your son was to pick 92 tiles, would he have a 2/92 chance of getting both jokers? No, he would have a 100% of getting both jokers. If he was picking 46 tiles (half of 92), then he would have a 25% of getting both jokers.

 

[Steven G]

Hi Dr Peterson thank you for your response. So when players pick up their 14 tiles the other players do not see each other’s tiles. My sons thoughts were that if the first person picks up 14 tiles then they have a 2/106 chance of getting a joker and when the second person picks up 14 tiles they then have a 2/92 chance because he felt like it was unlikely that the first person would have picked up a joker.

Probability is exact. When your son says that he felt like it was unlikely that the first person would have picked up a joker does not equate to the first person has 0 chance of picking up a joker. The 1st player has a chance of getting 1 or 2 jokers even those it might be unlikely and it will certainly change the chances that the 2nd player gets both jokers. Note that if the 1st player picked up 1 or both jokers, then the 2nd player has no chance at all of getting both jokers.

 

[Dr.Peterson]

Hi Dr Peterson thank you for your response. So when players pick up their 14 tiles the other players do not see each other’s tiles. My sons thoughts were that if the first person picks up 14 tiles then they have a 2/106 chance of getting a joker and when the second person picks up 14 tiles they then have a 2/92 chance because he felt like it was unlikely that the first person would have picked up a joker.

That would be the probability that the first tile they pick will be a joker. There will be a considerably bigger chance that there will be at least one joker among their 14.

A similar, but significantly simpler, problem can be found here:

The Cup Problem

Hello, I am currently trying to figure out the different probabilities of winning in an activity I completed. There are four cups with one cup having money underneath it. If four people chose a cup, one after another, what is the probability that each person would choose the cup with the money...

www.freemathhelp.com

This shows both our quick way to see that the order doesn't matter, and an explicit calculation. To do the latter for your question, you'd need to learn about combinations and conditional probabilities.