Probability question

**NB10** · 09-02-2019, 12:50 PM

Hey guys, I have a probability puzzle I'm really struggling to work out and was hoping someone here could help.

Two players roll one die each. Player 1 gets to add 2 to whatever number he rolls. High roll wins, and ties are re-rolled. What is the % probability that player 1 will win?

**Freightman** · 09-02-2019, 01:25 PM

When he rolls A 6, he will win 6 of 6
When he rolls a 5, he will also win 6 of 6
When he rolls a 4, he will win 5 of 6 and tie 1
When he rolls a 3, he will win 4 of 6, tie 1 of 6 and lose 1
When he rolls a 2, he will win 3 of 6, tie 1 of 6, and lose 2 of 6
When he rolls 1, he will win 2 of 6, tie 1 of 6, lose 3 of 6

The sum of his wins are 26
The sum of his ties are 4
The sum of his losses are 6

The probability of his winning any particular toss is is 26 of 36 is 13 of 18
The probability of his winning any subsequent toss after a tie, is still 13 if 18, however, his overall win odds will change slightly for each and every subsequent toss after an initial tie, and ties thereafter.

**NB10** · 09-02-2019, 01:35 PM

Originally Posted by Freightman

When he rolls A 6, he will win 6 of 6
When he rolls a 5, he will also win 6 of 6
When he rolls a 4, he will win 5 of 6 and tie 1
When he rolls a 3, he will win 4 of 6, tie 1 of 6 and lose 1
When he rolls a 2, he will win 3 of 6, tie 1 of 6, and lose 2 of 6
When he rolls 1, he will win 2 of 6, tie 1 of 6, lose 3 of 6

The sum of his wins are 26
The sum of his ties are 4
The sum of his losses are 6

The probability of his winning any particular toss is is 26 of 36 is 13 of 18
The probability of his winning any subsequent toss after a tie, is still 13 if 18, however, his overall win odds will change slightly for each and every subsequent toss after an initial tie, and ties thereafter.

Is there any way to incorporate the ties into one single calculation to get one % probability? I have no idea, this isn't a subject I've ever been able to understand fully.

**Freightman** · 09-02-2019, 01:38 PM

Originally Posted by NB10

Is there any way to incorporate the ties into one single calculation to get one % probability? I have no idea, this isn't a subject I've ever been able to understand fully.

It already is incorporated. For one toss, he will win 26 of 36 tosses, or 72.2% of the time.

**NB10** · 09-02-2019, 02:39 PM

Originally Posted by Freightman

It already is incorporated. For one toss, he will win 26 of 36 tosses, or 72.2% of the time.

Ah ok, see, I told you I don't get this stuff

Those are pretty good odds, thanks Freightman!

**ericfarmer** · 09-02-2019, 02:57 PM

Originally Posted by Freightman

It already is incorporated. For one toss, he will win 26 of 36 tosses, or 72.2% of the time.

This is incorrect, or at least misleading. The probability that player 1 wins is exactly 26/32=0.8125. That is, viewing a "round" as a sequence of dice rolls that continues until a player wins outright (i.e., re-rolling after a tie), player 1 will win 81.25% of rounds played.

**Freightman** · 09-02-2019, 04:18 PM

Originally Posted by ericfarmer

This is incorrect, or at least misleading. The probability that player 1 wins is exactly 26/32=0.8125. That is, viewing a "round" as a sequence of dice rolls that continues until a player wins outright (i.e., re-rolling after a tie), player 1 will win 81.25% of rounds played.

Semantics are semantics.

There is a caveat for 1 toss (post 4), which is covered by last paragraph of post 2. What you’re responding is my answer to a secondary question - both of our answers being both right or wrong - depending on how the question is interpreted. .

**ericfarmer** · 09-02-2019, 06:06 PM

Originally Posted by Freightman

Semantics are semantics.

There is a caveat for 1 toss (post 4), which is covered by last paragraph of post 2. What you’re responding is my answer to a secondary question - both of our answers being both right or wrong - depending on how the question is interpreted. .

"High roll wins, and ties are re-rolled" [my emphasis]. I'll let the OP clarify as needed, but this seems reasonably unambiguous to me.

But, now that we're in the discussion, your comment in your initial post that "his overall win odds will change slightly for each and every subsequent toss after an initial tie, and ties thereafter" is also incorrect. If I am asked to assess my probability of winning before we begin rolling, I will respond with 0.8125. If we roll and tie, and I am asked to re-assess my probability of winning (i.e., updated with the information that we just rolled and tied), I will again respond with 0.8125.

**Freightman** · 09-02-2019, 06:22 PM

Originally Posted by ericfarmer

"High roll wins, and ties are re-rolled" [my emphasis]. I'll let the OP clarify as needed, but this seems reasonably unambiguous to me.

But, now that we're in the discussion, your comment in your initial post that "his overall win odds will change slightly for each and every subsequent toss after an initial tie, and ties thereafter" is also incorrect. If I am asked to assess my probability of winning before we begin rolling, I will respond with 0.8125. If we roll and tie, and I am asked to re-assess my probability of winning (i.e., updated with the information that we just rolled and tied), I will again respond with 0.8125.

Regardless, semantics aside, I liked your version. Different wording certainly explains things in different ways.

**DSchles** · 09-02-2019, 06:52 PM

It would have been easier, in my view, and clearer, as well, as often is the case with BJ discussions, to state upfront that ties simply do not count, and that you continue to roll until there is a winner; i.e., you consider only resolved games.

That is clearly the intent of the question, but it might have been stated slightly differently. In any event, I agree with Eric; 26/32 is clearly the only correct answer.

Don