A little experiment. Please don’t analyse the statistics, and if you know the stats go with what you would have put.

Read the rules of the game, then make an instant decision on which seems best/gut instinct. Don’t try and outguess what I want please.

This is the game. It is played over 32 rounds, who wins the most rounds wins .

We each toss some coins (assume they are perfectly fair 50/50 coins). Who ever flips the most heads wins the round. Draws draw the round – it still counts as a round.

Your choice is toss 3, and reflip 1 (UPDATE -if you wish to, you only have to reflip a tail),

OR

toss 4.

What ever you choose must be used for all 32 flips.

[Note this was trapped as Spam, and I din’t see it until after I wrote the comment below – LH]

If we’re allowed to do the maths after we’ve voted:

4 tosses gives an expectation of 2 heads.

3 tosses gives an expectation of 1.5 heads.
The additional toss gives an expectation of an additional 0.5 heads, but only in the case where the inital result was not three heads, so the expectation is in fact 0.5 x (1-0.5^3).

Not only is the expectation lower for the 3+1 option, but the best possible score is lower, as the total number of heads can never exceed three. One time in every sixteen, the four coins will come up all heads.

This arose from a discussion on the Black Powder wargame Yahoo group.

One member was looking at how to model what he understands is the higher rate of fire of the Prussians during the Seven Years War, and he felt BP SYW Prussians should gain a fire advantage (note I hold no opinion on this) beyond the first round of firing.

The discussion was between 4 firing dice, and 3 plus a re-roll of any miss (the “sharpshooter” rule). In BP a roll of 4+ is a hit. I simplified this into another 50/50 for purposes of this post.

My own ‘gut instinct’ was there was no difference between the two in over all effectiveness, but the reroll would narrow the bell curve – less 0 hits balancing the possible 4 hits.

I produced a quick ‘Truth Table’ and saw that I was wrong.

There are 16 combinations 4 dice can roll – anyone who understands binary counting will see this – though many of these combinations will have identical effects: Miss Miss Hit Hit (MMHH) is effectively the same as HHMM. Count up the total number of hits- 32- and divide by 16 (combinations) and you get an average of 2 hits per throw – ie if your units have 4 fire dice, and between them roll 100 attacks, they will do, on average 200 hits.

So far, so obvious.

The 3+a reroll gives an average of 1.94 hits. Ok you can’t get .94 of a hit, but over the course of the same 100 shots they will cause, on average 194 hits- meaning 4 dice have an advantage of 3% (for comparison the ‘house’ advantage in roulette is 2.7%, and that makes money!)

I immediately realised my mistake – the reroll (or reflip in the game above) is effectively your 4th die, EXCEPT as there is no miss/tail if you roll 3 hits to start with, then you do not get the 4th chance. The distribution is exactly the same as for 4 dice, except there isn’t the 1 in 16 chance of hitting HHHH. The table for 3 dice is 8 combinations, so in 16 flips (to match the 16 combos for 4 dice) you would get the same spread twice, then the rerolls – half of which will be a ‘bonus’ hit, the other half not EXCEPT the HHH can’t get the 4th hit, so combo 32 is still only HHH (plus effectively a miss).
[NB- this section has been rewritten after I doubted my self THEN came up with the wrong answer – I have cvhanged it back to this now. Thanks to Dom Skelton for pointing this out.]

I commented on the Yahoo group that I wonder how many people just saw the reroll, and assumed they would be less likely to roll no hits. In reality the chance of 3 misses on 3 dice is 1 in 8 (50% chance x 3 rolls = 12.5%), but they could still roll a miss on the reroll, which is the same as rolling 4 misses on 4 dice!

This is the reason I set up this little experiment – how many people would automatically go “Reroll! That’s a chance to improve the 4 dice don’t get”. However in the above rules you are more likely to win if you take 4 dice – you are likely to roll evenly between you, except every so often you will get an unbeatable 4 hits.

Ok, so over 32 rolls (chosen as 2 x 16) you may find your only 4 was matched to his only 0, and you lost everything else. However the more rounds you play, or the more games are played, the more statistics will favour you. (I realise it is possible to go 32 rounds with out a win on 4 coins – 128 tails, but it is highly unlikely 1 in
340,282,366,920,938,463,463,374,607,430,000,000,000 – if you tosses all 128 coins once every second then there would be a 50% chance of it happening after 5,391,596,376,141,982,229,457,675,968,639 years. The universe is 13,700,000,000 years old)

of the 76 responses I got (to this point)
41 (54%) took the reroll
35 (46%) took 4 dice

Many (most?) of these were wargamers, as I linked here from The Miniatures Page, and they (should) have a passing aquaintence with statistics of dice throwing! However, it appeals the human mind is too easily tricked by the promise of ‘something extra’.

If you got the wrong answer please feel free to explain why (likewise if you picked correctly for the wrong reason!)

[Note this was trapped as Spam, and I din’t see it until after I wrote the comment below – LH]

If we’re allowed to do the maths after we’ve voted:

4 tosses gives an expectation of 2 heads.

3 tosses gives an expectation of 1.5 heads.

The additional toss gives an expectation of an additional 0.5 heads, but only in the case where the inital result was not three heads, so the expectation is in fact 0.5 x (1-0.5^3).

Not only is the expectation lower for the 3+1 option, but the best possible score is lower, as the total number of heads can never exceed three. One time in every sixteen, the four coins will come up all heads.

THE EXPLANATION

This arose from a discussion on the Black Powder wargame Yahoo group.

One member was looking at how to model what he understands is the higher rate of fire of the Prussians during the Seven Years War, and he felt BP SYW Prussians should gain a fire advantage (note I hold no opinion on this) beyond the first round of firing.

The discussion was between 4 firing dice, and 3 plus a re-roll of any miss (the “sharpshooter” rule). In BP a roll of 4+ is a hit. I simplified this into another 50/50 for purposes of this post.

My own ‘gut instinct’ was there was no difference between the two in over all effectiveness, but the reroll would narrow the bell curve – less 0 hits balancing the possible 4 hits.

I produced a quick ‘Truth Table’ and saw that I was wrong.

There are 16 combinations 4 dice can roll – anyone who understands binary counting will see this – though many of these combinations will have identical effects: Miss Miss Hit Hit (MMHH) is effectively the same as HHMM. Count up the total number of hits- 32- and divide by 16 (combinations) and you get an average of 2 hits per throw – ie if your units have 4 fire dice, and between them roll 100 attacks, they will do, on average 200 hits.

So far, so obvious.

The 3+a reroll gives an average of 1.94 hits. Ok you can’t get .94 of a hit, but over the course of the same 100 shots they will cause, on average 194 hits- meaning 4 dice have an advantage of 3% (for comparison the ‘house’ advantage in roulette is 2.7%, and that makes money!)

I immediately realised my mistake – the reroll (or reflip in the game above) is effectively your 4th die, EXCEPT as there is no miss/tail if you roll 3 hits to start with, then you do not get the 4th chance. The distribution is exactly the same as for 4 dice, except there isn’t the 1 in 16 chance of hitting HHHH. The table for 3 dice is 8 combinations, so in 16 flips (to match the 16 combos for 4 dice) you would get the same spread twice, then the rerolls – half of which will be a ‘bonus’ hit, the other half not EXCEPT the HHH can’t get the 4th hit, so combo 32 is still only HHH (plus effectively a miss).

[NB- this section has been rewritten after I doubted my self THEN came up with the wrong answer – I have cvhanged it back to this now. Thanks to Dom Skelton for pointing this out.]

I commented on the Yahoo group that I wonder how many people just saw the reroll, and assumed they would be less likely to roll no hits. In reality the chance of 3 misses on 3 dice is 1 in 8 (50% chance x 3 rolls = 12.5%), but they could still roll a miss on the reroll, which is the same as rolling 4 misses on 4 dice!

This is the reason I set up this little experiment – how many people would automatically go “Reroll! That’s a chance to improve the 4 dice don’t get”. However in the above rules you are more likely to win if you take 4 dice – you are likely to roll evenly between you, except every so often you will get an unbeatable 4 hits.

Ok, so over 32 rolls (chosen as 2 x 16) you may find your only 4 was matched to his only 0, and you lost everything else. However the more rounds you play, or the more games are played, the more statistics will favour you. (I realise it is possible to go 32 rounds with out a win on 4 coins – 128 tails, but it is highly unlikely 1 in

340,282,366,920,938,463,463,374,607,430,000,000,000 – if you tosses all 128 coins once every second then there would be a 50% chance of it happening after 5,391,596,376,141,982,229,457,675,968,639 years. The universe is 13,700,000,000 years old)

of the 76 responses I got (to this point)

41 (54%) took the reroll

35 (46%) took 4 dice

Many (most?) of these were wargamers, as I linked here from The Miniatures Page, and they (should) have a passing aquaintence with statistics of dice throwing! However, it appeals the human mind is too easily tricked by the promise of ‘something extra’.

If you got the wrong answer please feel free to explain why (likewise if you picked correctly for the wrong reason!)

The count now stands at 45 vs 37 – still the same proportion.