> It is fair to ask why the likelihoods are useful if they are so small
The way the question demonstrates "smallness" is wrong, however. They quote the product of the likelihoods of 50 randomly sampled values - 9.183016e-65 - as if the smallness of this value is significant or meant anything at all. Forget the issue of continuous sampling from a normal distribution, and just consider the simple discrete case of flipping a coin. The combined probability of any permutation of 50 flips is 0.5 ^ 50, a really small number. That's because the probability is, in fact, really small!
The way the question demonstrates "smallness" is wrong, however. They quote the product of the likelihoods of 50 randomly sampled values - 9.183016e-65 - as if the smallness of this value is significant or meant anything at all. Forget the issue of continuous sampling from a normal distribution, and just consider the simple discrete case of flipping a coin. The combined probability of any permutation of 50 flips is 0.5 ^ 50, a really small number. That's because the probability is, in fact, really small!