# Measurement of probability

#### samr

##### New Member
Suppose one throws a coin 100 times, and gets exactly 50 heads. This is precisely the expected number of heads on average. Nevertheless, if one calculates the probability of 50 successes out of 100 trials, with 0.5 being the probability of success, the probability is :
(100 over 50) * 0.5^50* 0.5^50 ~= 0.08.

This is not a high probability, but I think (perhaps wrongly) that the fact that the result is equal to the expectation, means that the result it is what we would call "probable" in daily life.

What test should one use to check whether such a result is indeed "probable"?

#### fed2

##### Active Member
the test would be to look at the number 8% and then decide how you feel about that in terns of your personal feelings of probable-ness.

#### Buckeye

##### Active Member
Suppose one throws a coin 100 times, and gets exactly 50 heads. This is precisely the expected number of heads on average. Nevertheless, if one calculates the probability of 50 successes out of 100 trials, with 0.5 being the probability of success, the probability is :
(100 over 50) * 0.5^50* 0.5^50 ~= 0.08.

This is not a high probability, but I think (perhaps wrongly) that the fact that the result is equal to the expectation, means that the result it is what we would call "probable" in daily life.

What test should one use to check whether such a result is indeed "probable"?
Maybe you are missing a summand in the formula for expectation? https://proofwiki.org/wiki/Expectation_of_Binomial_Distribution

This is not a high probability, but I think (perhaps wrongly) that the fact that the result is equal to the expectation, means that the result it is what we would call "probable" in daily life.

What test should one use to check whether such a result is indeed "probable"?
I'm not quite sure what you mean here. But, the law of large numbers might help fill in gaps. Or maximum likelihood: https://online.stat.psu.edu/stat504/lesson/1/1.5

Last edited:

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Not my area but 50% at 8% of time seems pretty great. Yes given the law of large numbers it should coverage to 50% as tosses approach infinity. I am guessing this is something like 8% mass right there and is is symmetrical on both sides.

#### AngleWyrm

##### Active Member
You could plot a chart showing P(k) for each of the possible k in the range 0..100 counts of successes out of 100 trials. The sum totals 100%

A grain of sand is not very big when compared to a mountain, but when compared to another grain of sand it's size might make more sense.

Last edited: