The probability of obtaining all heads or exactly fifty heads in 100 tosses of a fair coin is (1/2)^100. While the probability for both scenarios is mathematically identical, the intuition surrounding the likelihood of specific sequences can be misleading. The discussion highlights the difference between obtaining a specific sequence versus the overall distribution of outcomes, emphasizing that longer runs of identical outcomes are statistically less probable. A computer simulation was conducted to analyze run lengths, reinforcing the concept that while specific sequences have equal probabilities, their occurrence can vary significantly in practice.
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How did you get that answer for the second case, getting 50 heads?jk22 said:Suppose I toss a fair coin 100 times. If I consider the order of apparition, then obtaining all head, or fifty times head has the same probability, namely (1/2)^100 ?
No, because there is only one way in which you can toss all heads, but there are many more ways in which you can obtain 50 heads in 100 tosses.jk22 said:Suppose I toss a fair coin 100 times. If I consider the order of apparition, then obtaining all head, or fifty times head has the same probability, namely (1/2)^100 ?
jk22 said:I'm considering one particular order hhhh...ttttt... fifty time each.
I'm a bit confused with your wording there, did you mean getting 50 consecutive heads followed by 50 consecutive tails and you want to know the chance of getting that particular order? If that's the case, it should be indeed (1/2)^100.jk22 said:I'm considering one particular order hhhh...ttttt... fifty time each.
Then you may want to refine your intuition, getting one particular order is as hard as getting another different particular order.jk22 said:But, intuitively, it seems to me that the serie htht... were more plausible than hhhh...ttt... ?
I think you may be confusing the statistics of run length and the maximum run length of a given outcome. On average, you will get many runs of length 1, but you have only one possibility (if you start by h) that the longest run is of length 1.jk22 said:But that's where my problem arises above, since run length that are longer have less chance to appear, I would expect the case hhhh...tttt... were less likely to appear.