The probability of 3 names begin with the same letter The names
Mary
Mario
Michelle
start with the same letters. which are the odds? I would guess $\left(\frac1{26}\right)^3$, but I'm not sure.
 A: Not quite. The probability that three names start with a given letter, in this case, $M$ is $(1/26)^{3}$, but you asked for the probability that three names start with the same letter.
Since there are $26$ choices for the letter, the probability that three names start with the first letter is $26\cdot(1/26)^{3}=(1/26)^{2}$.
A: The answer depends on the distribution of the first letters of given names (weighted by their frequency in the population). If for concreteness we assume that we're using the standard English Roman alphabet $\Sigma := \{A, B, C, \ldots\}$ and we denote the fraction of the population whose names begin with a given letter respectively by $P_A, P_B, P_C, \ldots$, then the odds that three randomly selected people all have first names starting with a letter $\star$ is $P_{\star}^3$. Adding over all $26$ letters gives that the probability that three randomly selected people all have the same first initial is
$$P = P_A^3 + \cdots + P_Z^3 = \sum_{\star \in \Sigma} P_{\star}^3 .$$
In the simple but unrealistic case that the first letters $\star$ are uniformly distributed, that is, that $P_A = \cdots = P_Z = \frac{1}{|\Sigma|} = \frac{1}{26}$, we have
$$\color{#bf0000}{\boxed{P = \sum_{\star \in \Sigma} \left(\frac{1}{26}\right)^3 = 26 \cdot \left(\frac{1}{26}\right)^3 = \left(\frac{1}{26}\right)^2 \approx 0.1479\%}} .$$ It's not too hard to show, though, that among all distributions the uniform distribution actually leads to the smallest probability $P$.
The probability using real-world data is often much larger: The distribution of the 90% most common names (which should be reasonably close to the distribution of all names) in the 2000 U.S. census is reported in an old Google Answers thread: $P_A = 0.06508$, $P_B = 0.04648$, $P_C = 0.07223$, ... . Computing using this data gives a good practical estimate for the U.S.:
$$\color{#bf0000}{\boxed{P = \sum_{\star \in \Sigma} P_{\star}^3 = 0.5808\% \approx \frac{1}{172}}} ,$$
which is about $4$ times the probability for a uniform distribution. (Interestingly, computing using data from a less-than-authoritative-looking source gives that the corresponding figure for last names is $0.4191\%$, rather smaller than the first name figure.) Some practical evidence suggests that the distribution of first names follows a power law distribution. 
