# Probability that multiple confidence intervals contain the true population mean

Suppose you take 3 samples and report a 99% confidence interval for each dataset. How would you go about calculating the probability that all the datasets contain the true population mean? No other statistics are provided.

• Presumably the samples are independent, right? – whuber Apr 6 '12 at 17:47
• Do you mean all intervals contain the true mean ? If the samples are independent then obviously the probability that the three intervals contain the mean is $0.99^3$. – Stéphane Laurent Apr 6 '12 at 19:00

Well what you could say is that if you repeated this experiment of generating three 99% confiodence intervals independently many times, the percentage of the time that the three intervals will all include the true mean is $100\times .99^3$ which is what Stephane said and is I think what the author intended.
I'll be careful with this question and suggest you to be careful with the answers. Recall that a $100\alpha\%$ confidence interval DOES NOT MEAN that there is a probability $\alpha$ of finding the true mean in the interval: remember that the true mean, $\mu$, is in the interval or it is not. This mean is not a random variable and, therefore, we cannot associate probabilities to it: the statistic, on the other hand, is random.