Let’s say you take three pennies and toss them up in
the air simultaneously. What are the odds they all come down on the same side,
either heads or tails?

Well, obviously, there’s a 50/50 chance a flipped coin will result in either a head or tail. A one in two chance, or 1/2.

For three coins, the probability of all landing all
the same would be

(1/2) x (1/2) x (1/2)

Which is 1/8. One out of eight times, or 12.5% of the
time.

But …

When you toss three coins up in the air, at least two
will come down the same, right? If they don’t all land either all heads or all
tails, at least two will be heads or tails.

So that leaves the third coin in question.

It has a 50/50 chance of agreeing with the other two
coins. A one in two chance, or 1/2.

Therefore, the odds of flipping three coins
simultaneously and getting three matches would be

(1/2)

Which is 1/2. One out of two times, or 50% of the
time.

So ….

Which is the correct probability? One-eighth of the
time or one-half?

Why?

## 8 comments:

i say 1/8 because the 3rd penny may be one of the pair that will both be heads or tails where the first 2 are already different.

Uncle

Actually, I'd like to change my vote. There are 8 permutations, however, 2 of the 8 are HHH or TTT. In that case the odds that all 3 end up the same are 1/4.

Uncle

I'm leaning towards the 1/8 all heads or tails (or 1/4 as either as my wording in the post wasn't as clear as should've been). But I'm susceptible to the second argument (though a nagging feeling tells me it's sophistry and I get a headache). I want to see our mathematician nephew/grandnephew weigh in....

I feel like there's situations of where the order of the coins matters or it doesn't, and whether you would like heads or tails or you simply don't care. Say we want to find out the probability of all three landing on heads, and flipping them in sequence. This case is simple seeing as the first coin has a 1/2 chance, the second has a 1/2 chance, and the third has a 1/2 chance. So we say it has a 1/8 possibility of happening. This is a permutation where repetition is allowed, the equation for the amount of possibilities being n^r (n being possible options, heads or tails, with r trials, n=2,r=3). Now the same can be said for the probability of three tails being flipped. If we don't care about it being heads or tails we can just add up the probability of each, 1/8 + 1/8 and get 1/4. We can arrive to this again by saying it doesn't matter what the first coin lands on, and having its probability of matching seeing as it has nothing to follow as equal to 1. The chance the second coin will match the first is 1/2, and the chance the third will match the second is 1/2. Here it can be seen as 1*(1/2)*(1/2) or 1/4.

Now for throwing all three up in the air simultaneously and not in sequence this changes a little bit. We can be 100% certain that at least two of the coins will land with the same side facing up. At this point it is simply a matter of if the third coin is the same as the other two and we have a probability of all three being heads or tails up as 1/2.

Now let’s consider if we throw up three coins and want two coins to initially match, ignoring if we want either heads or tails . Lets label the coins A,B,C. If you throw up all three coins and you want coin A to be the same as coin B you have a 1/2 chance of that happening. The chance now that C will be the same as A is 1/2. Here we see the chance of all three being the same as (1/2)*(1/2)=1/4. The same can be said for any case where you want a specific two coins to match each other. This can also be determined by making a simple 2x2 matrix almost like a punnett square. The first row is A=B, the second row is A!=B, the first column is C=A, and the second column is C!=A. Put a check in a box where all coins are equal and an x where they are not and you will see only 1/4 boxes show that scenario.

Now let’s consider when we want a specific two coins to match out of the three and we want the a specific side of the coin. Let’s say we want heads, and coins A and B to match. The chance of A being heads is 1/2. The chance that B will be the same as A is 1/2, and the chance that C will be the same as A is 1/2. We now conclude that the possibility of all this case happening is 1/8.

In summary it comes down to whether you want heads or tails, and if you want specific coins to be the same. Flipping 3 coins in sequence with a preferred coin side facing up is a 1/8 chance, flipping 3 coins in sequence and having them all land the same side up is a 1/4 chance. Throwing three coins up and having them just land all on the same side up is a 1/2 chance. Throwing three coins up and having a specific two be the same with no desired side is a 1/4 chance, and throwing up three with two being the same side and a desired side is a 1/8 chance.

Nephew

So, since sequence/order doesn't matter in the question as stated, the answer is 1/2 - a 50% chance that three coins simultaneously will land on the same side. I understand that there is a 1/8 probability of hitting three heads in a row on a single coin tossed three times. It gets murky with the simultaneity. Plus the fact that specificity is not important. Have to google punnett squares to fully follow that argument though.

And .... I'll be spending half of Christmas Eve 2017 flipping three pennies in the air and logging my results. :( My in-laws are gonna think I'm weirder than they've long ago realized I am ...

If I throw 3 coins in the air at the same time, the results will be one of

HHH

HTT

HTH

HHT

TTT

THH

THT

TTH

As such, it is still 1/4 that they will land HHH or TTT. To me its only 1/2 if you know the first 2 landed the same way.

Uncle

This settled it for me and goes back to my very first comment that you cant assume THE first 2 come down the same. 1/4.

As nicely explained by Francis Galton himself in his very readable paper in Nature (1894), the fallacy lies in confusing a particular coin with any coin. The argument goes:

1) At least 2 of the coins must turn up alike.

2) It is an even chance whether A third coin is heads or tails

3) Therefore, it is an even chance whether THE third coin is heads or tails.

Wrong! 'A third coin'; is not the same as 'the third coin'!

Looking at the 8 possible outcomes, you can see that there are 4 ways for any 2 coins of the 3 to be both heads: HHH, HHT, HTH, and THH. But the third coin is, respectively, H,T,T,T - that is, there is only a 1/4 chance - not 1/2 - that the third coin is the same as the 2 coins that agree! That's because the third coin is actually not one particular coin, but may be any of the three.

Uncle

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