note that the order of the flips is authoritative if we want to ensure our results are evenly distributed—HT is not the same solution as TH. With this in heed, if we want a 25 % probability, we choose TT ( two tails in a row ) as our “ win ” condition and then flip our single coin doubly. We will get TT with 25 % probability and a unlike result with 75 % probability .
however, if we want a probability result that ‘s a little more exotic—say, one-third—then we have a snatch of a problem. It ‘s not immediately obvious how to get a 13\frac1331 probability out of a coin. We need three outcomes with equal probability if we ‘re going to get a one-third probability, but a one flip of the mint only gives us two, and two flips of the coin give us four.
Two flips of the coin give us besides many results. ( We can see the four potential outcomes above : HH, HT, TH, TT. ) however, since we want an exact one-third probability, let ‘s just decide before we begin that if we flip HH, we ‘ll throw out the consequence and depart over ; now we ‘re down to three outcomes !
With this approach path in mind, we can lay out the results of our flip visually. Let ‘s agree that TT is a “ gain ”, both HT and TH will count as a “ loss ” each, and HH will be thrown out and require begin over. then we have the follow :
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The difficulty here is how to think about the grey area at the top. It represents the HH that we decided we ‘d ignore—when we get two heads, we have to start over. But wait ! Since we ‘re starting over, the raw set of flips will have the like probability distribution as our first try. We can replace the grey area with the master triangle ( repeatedly ) like this :
What is the helping of the entire area that is purple ? It feels a little daunt to try to calculate the imperial area in this prototype ( so many triangles ! ). however, there ‘s a way to find the answer without calculation.
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note that we can think of this complicate determine as a series of rows. Start at the penetrate and look at the three largest triangles. Each row of equal-sized triangles has three shapes : two orange triangles and one purple triangle, all of equal area. frankincense, the purple is precisely one-third of each row. Since the entire shape can be turned into these rows, the purple is precisely one-third of the total area ! therefore, the probability of getting two tails in a row is precisely one-third, equally long as we decide ahead of time that we ‘re going to start over whenever we get two heads .
It turns out that we can get precisely one-third probability from a one carnival mint. What early probability distributions are possible ?