# Calculating observed probability

#### ThomasCNot

##### New Member
Can anyone point me towards the formula and extra reading I would need to calculate how many observations I would need to make in order to calculate an observed probability within a set confidence interval?

So for example, to calcuate how many times would I need to toss a coin to observe that p(heads)=0.5, within 95%CI?

Thank you,

Thomas

#### JesperHP

##### TS Contributor
So for example, to calcuate how many times would I need to toss a coin to observe that p(heads)=0.5, within 95%CI?
You can set up the interval of a 95% confidence interval for any sample size, the question is how wide it is. So assuming the confidence interval so be smaller than L what should the sample size be? Well given that you are using the simple sample average as estimator and
you assume it is approximately standard normal the 95% confidence interval +/- 1.96 standard errors. So the total length is about 4 standard errors and the standard error is sqrt(p(1-p)/n).... p(1-p) is maximized for p=0.5 ... so if n is chosen large enough to ensure an interval shorter than L for p=0.5 it will also hold when p is not 0.5.

#### ThomasCNot

##### New Member
Thank you for your reply. I looked through the article you suggested. Would the following be a sensible approach:

1. Take some observations, say 100 to get a feel for what p might be (observations for me are cheap).
2. Let's say it looks like p=0.1. Using the formula, calculate n when 4*sqrt(p(1-p)/n)=W units wide interval. So then all I'd have to do is choose an acceptable interval.

#### JesperHP

##### TS Contributor
Sounds reasonable ... however if you measure $$0.1=\hat p$$ based on 100 obs. perhaps you end up with $$0.15=\hat p$$ for a larger sample and a wider confidence interval than what you wanted... you could simply add a little to the $$\hat p$$ measured with 100 obs in order to add some conservatism ...

heres a table of standard errors sqrt(p(1-p)/n)*100 for different sample sizes - columns - and different p's - rows. The standard errors are multiplied with 100 so they are en percentage points.

Code:
          100      500      1000      1500      2000      2500      3000      3500      4000      4500      5000
0.1  3.000000 1.341641 0.9486833 0.7745967 0.6708204 0.6000000 0.5477226 0.5070926 0.4743416 0.4472136 0.4242641
0.15 3.570714 1.596872 1.1291590 0.9219544 0.7984360 0.7141428 0.6519202 0.6035609 0.5645795 0.5322906 0.5049752
0.2  4.000000 1.788854 1.2649111 1.0327956 0.8944272 0.8000000 0.7302967 0.6761234 0.6324555 0.5962848 0.5656854
0.25 4.330127 1.936492 1.3693064 1.1180340 0.9682458 0.8660254 0.7905694 0.7319251 0.6846532 0.6454972 0.6123724
0.3  4.582576 2.049390 1.4491377 1.1832160 1.0246951 0.9165151 0.8366600 0.7745967 0.7245688 0.6831301 0.6480741
0.35 4.769696 2.133073 1.5083103 1.2315302 1.0665365 0.9539392 0.8708234 0.8062258 0.7541552 0.7110243 0.6745369
0.4  4.898979 2.190890 1.5491933 1.2649111 1.0954451 0.9797959 0.8944272 0.8280787 0.7745967 0.7302967 0.6928203
0.45 4.974937 2.224860 1.5732133 1.2845233 1.1124298 0.9949874 0.9082951 0.8409179 0.7866066 0.7416198 0.7035624
0.5  5.000000 2.236068 1.5811388 1.2909944 1.1180340 1.0000000 0.9128709 0.8451543 0.7905694 0.7453560 0.7071068
0.55 4.974937 2.224860 1.5732133 1.2845233 1.1124298 0.9949874 0.9082951 0.8409179 0.7866066 0.7416198 0.7035624
0.6  4.898979 2.190890 1.5491933 1.2649111 1.0954451 0.9797959 0.8944272 0.8280787 0.7745967 0.7302967 0.6928203
0.65 4.769696 2.133073 1.5083103 1.2315302 1.0665365 0.9539392 0.8708234 0.8062258 0.7541552 0.7110243 0.6745369
0.7  4.582576 2.049390 1.4491377 1.1832160 1.0246951 0.9165151 0.8366600 0.7745967 0.7245688 0.6831301 0.6480741
0.75 4.330127 1.936492 1.3693064 1.1180340 0.9682458 0.8660254 0.7905694 0.7319251 0.6846532 0.6454972 0.6123724
0.8  4.000000 1.788854 1.2649111 1.0327956 0.8944272 0.8000000 0.7302967 0.6761234 0.6324555 0.5962848 0.5656854
0.85 3.570714 1.596872 1.1291590 0.9219544 0.7984360 0.7141428 0.6519202 0.6035609 0.5645795 0.5322906 0.5049752
0.9  3.000000 1.341641 0.9486833 0.7745967 0.6708204 0.6000000 0.5477226 0.5070926 0.4743416 0.4472136 0.4242641