# Given we observe $x$ successes in sample of $n$, what is the probability that $p>0.5$

I flip a coin $$n$$ times, and I find that for $$x$$ of these it lands on heads. Let $$p$$ be the true probability of the coin landing on heads, though this is currently unknown.

I am interested in $$P(p>0.5 \mid$$ I flipped it $$n$$ times seeing $$x$$ occurrences of heads$$)$$.

I would really appreciate if someone could help me with:

(a) Working this probability out

(b) If there's a function in R I could use to do this (inputting the $$n$$ and $$x$$ I have for the different coins).

• You need to supply at least one more piece of information: what do you currently think $p$ is and how strongly do you believe that? This information is usually encoded in a prior distribution. If you don't have such a distribution in mind, then you are probably asking the wrong question: instead, you might be looking for a hypothesis test.
– whuber
May 20, 2020 at 15:09
• Thanks for the quick reply! I didn't have an expected value of $p$ in mind, all I'm interested in is given I've seen $x$ out of $n$ successes, how likely it is that the value of $p$ is greater than $0.5$. I'm not sure I understand why I'm not able to work out such a probability from the information I provided. For example if I observed $n = x = 100$, I'd expect the probability that the 'true value of $p$ is larger than $0.5$' to be fairly high (though I have no idea how to actually calculate it). May 20, 2020 at 15:23
• Well, you have observed $p'=\frac{x}{n}$ successes. If $p'>p$, your probability is 1, else it is 0. Either way, you are done. Does this answer your question? If not, your counter-argument is the beginning of the discussion of why you need a prior probability. May 20, 2020 at 15:27
• Sounds like I need to read up about prior probabilities some more! For now though, given I have no further context around whether this coin is biased, the sensible thing to do seems to be to take the prior distribution to be binomial with $p=0.5$. I'll update the question to reflect this! May 20, 2020 at 15:42

Well the probability that you are asking for is not computable, at least not using frequentist statistics. Observe that in general, a conditional probability is computed as

$$P(A|B)=\dfrac{P(A\cap B)}{P(B)}$$

But here, $$P(B)=$$ I flipped the coin n times seeing x occurrences of heads. And for this probability we would need to know the value of $$p$$.

You can, as sugested in the comments, consider a Bayesian perspective, but for doing that you would require knowledge on a prior distribution.

What you can actually do from a frequentist perspective is to comupte a hypothesis test. For doing this, you should define your null hypothesis: $$H_0: p=0.5$$

And the alternative hypothesis , which is:

$$H_1:p>0.5$$

Now we can compute the test statistic

$$T = \frac{\hat{p}-p_{0}}{\sqrt{p_{0}\left(1-p_{0}\right) / n}} \sim_{a p} N(0,1)$$

where $$p_0=0.5$$ (the value under your null hypothesis) and $$\hat{p}=x/n$$ is the sample proportion. plugging in the values for $$p_0$$ annd $$\hat{p}$$, T is an actual number.

Now consider a significance level $$\alpha = 0.05$$ for your test. Then you can check using for example, the program R, that $$z_{\alpha}=1.64$$. This means that $$P(N(0,1)>1.64) = 0.05$$

Finally, assuming that the null hypothesis is true, T is drawn approximately from a normal distribution, so you can compare T against the value you would expect from a normal distribution, which is 1.64

• If $$T>1.64$$ then T is a strange value, because it has a probability of being from a normal distribution smaller than 5%. You reached this conclusion assuming that the null hypothesis was true, so you conclude that the null hypothesis should be rejected and the alternative hypothesis should be supported.

• If $$<=1.64$$ you cannot reject the null hypothesis.

Doing this in R:

n = 100 # number of tosses
x = 60 # number of heads
p = 0.5 # your null hypothesis probability

binom.test(x=x, n=n, p=p, alternative='greater', conf.level=0.95)


Which returns

Exact binomial test

data:  x and n
number of successes = 60, number of trials = 100, p-value = 0.02844
alternative hypothesis: true probability of success is greater than 0.5
95 percent confidence interval:
0.5129758 1.0000000
sample estimates:
probability of success
0.6


So the p-value here is $$0.02844$$. If the p-value is smaller than $$\alpha=0.05$$ then you do not reject the null hypotheis. If the p-value is greater than $$\alpha$$ then you reject the null hypothesis