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Assume that we flip a coin that has a probability of $p > 0.5$ to come up heads.

If we flip the coin $n$ times, we can define the binomial tail as the probability of observing less than or equal to $k$ heads. Let's define that as: $$P(n,k,p) = \sum_{i = 0}^k {n \choose i} p^i (1-p)^{n-i}$$

Is it true that $P(n,\frac{n}{2},p) > P(n+1,\frac{n+1}{2},p)$ for $p > 0.5$? Or with other words, does the probability of observing more tails than heads decrease if we flip the coin more often?

Intuitively, this should be true, since the more we flip the coin, the more representative the sample must be. Thus, observing less heads than tails should become more unlikely, since $p>0.5$. But I can't see how to prove this.

Note that there are some issues with rounding with this question, which I'm not super sure how to resolve elegantly. Any suggestions are most welcome.

Tom
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  • Where does rounding occur in this circumstance? You don't seem to be asking about numerical computation. Your question is readily answered by considering $p=0.5+\epsilon$ for tiny $\epsilon$ and looking at the cases $n=1,2,3.$ There's an issue with the difference between odd and even $n:$ the parity of $n$ affects whether the value of $n/2$ is included in the calculation of $P(n,n/2,p).$ You would be better off comparing $P(n,n/2,p)$ to $P(n+2,n/2+1,p).$ – whuber Feb 20 '22 at 19:35
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    If $n$ is odd, $\frac{n}{2}$ is not an integer and the definition of $P(n,k,p)$ doesn't work anymore. Similairly, $\frac{n+1}{2}$ is not an integer if $n$ is even. – Tom Feb 20 '22 at 19:37
  • $P$ is well-defined even when $n/2$ is non-integral. Try it out in R, for instance. Here's a plot of the first 30 values (for a fixed $p$): P <- function(n, p) pbinom(n/2, n, p); plot(P(1:30, 1/2+1e-6)) – whuber Feb 20 '22 at 19:38
  • BTW, https://stats.stackexchange.com/questions/561978 provides excellent insight. It implies that if you modify the Binomial CDF to include one-half the probability at each jump (rather than the full probability), then half the time the two probabilities in your question are equal (no matter what value $p$ might have!) and the other half of the time they indeed decrease (for $p\ge 1/2$ only). – whuber Feb 20 '22 at 20:00

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