In the previous video, we worked through the RU-486 example with four successes, pregnancies, in 20 trials, subjects. Our prior distribution was uniform, except that we placed a higher probability mass at p equals 0.50. The likelihood of the data was calculated using the binomial distribution. And peaked at p is equal to 0.20. Putting together the prior and the likelihood, we obtained a posterior distribution with the highest posterior probability at p is equal to 0.20. While there's a peak at this value, there's still some uncertainty in the posterior, as other models also have some probability mass in the posterior distribution. Next, let's take a look at what the posterior distribution would look like if we had more data. Suppose our sample size was 40 instead of 20, and the number of successes was 8 instead of 4. Note that we're still maintaining the 20% ratio between the sample size and the number of successes. We'll start with the same prior distribution. Calculate the likelihood of the data which is also centered at 0.20, but is less variable than the original likelihood we had with the smaller sample size. And finally put these two together to obtain the posterior distribution. The posterior also has a peak at p is equal to 0.20, but the peak is taller. In other words, there is more mass on that model, and less on the others. To illustrate the effect of the sample size even further, we're going to keep increasing our sample size. Still keeping the 20% ratio between the sample size and the number of successes. So let's consider a sample with 200 observations and 40 successes. Once again, we're going to use the same prior and the likelihood is again centered at 20% and almost all of the probability mass in the posterior is at p is equal to 0.20. The other models do not have 0 probability mass, but they're posterior probabilities are very close to zero. So as you can see, as more data are collected, the likelihood ends up dominating the prior. This is why, while a good prior helps, a bad prior can be overcome with a large sample. However, it's important to note that this will only work as long as we don't place a zero probability mass on any of the models in the prior.