I have following formula ("n" is number of experiments, "k" is number of successes that we expect, "p" is probability of success, "q" is probability of failure):
I used it to calculate chances of getting strictly 2 heads out of 4 tosses (order doesn't matter), 5 heads of 10 tosses, 10 heads out of 20 tosses.
2 heads out of 4 tosses have around 38% probability.
5 heads out of 10 tosses have around 25% probability.
10 heads out of 20 tosses have around 18% probability.
Looks like trend for me. And this trend is paradoxical, it means that by increasing number of tries we will decrease chance for 50/50 outcome. It seems to be at odds with what I was taught before. I was taught that as we increase number of tosses we will come closer and closer to 50/50 outcome. So if I toss a coin 10 times I can get 1 head and 9 tails, while tossing a coin 1000 times is more likely to give me result much closer to 50/50 outcome.
Best Answer
This is because the more you flip a coin, the less likely it will be that you get heads $exactly$ half of the time. It is true, however, that the more you flip a coin, and then compute
$$\hat{p}=\frac{\text{number of heads}}{\text{number of tosses}}$$
that $\hat{p}$ will $converge$ to $0.5$. Consider flipping a coin $10$ times and obtaining $6$ heads. Then consider flipping a coin $1000$ times and obtaining $526$ heads.
While in the second instance, we are $26$ heads away from a perfect $50/50$, proportionally, we are closer as $0.526$ is closer to $0.5$ than $0.6$.