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http://en.wikipedia.org/wiki/Slash_distribution It should be possible to derive the probability $<1/M$ from there. … The proportion of cases in which it is true would be the probability of your event. …

This is the linear regression coefficient in the simple linear model. If $$y_i=a+bx_i$$ with $i=1,...,n$, then $$b=cov(Y,X)/var(X).$$ Similarly, the Pearson correlation is defined as $$cov(Y,X)/(var(X …

This probability is very small, so you can say with very low probability of error that observing 33 unique of 44 words was not caused by chance, because if the null hypothesis was true the pobability of … Put differently, position seems to have an impact on the probability of uniqueness. Only in a very small proportion of cases you would make an error when claiming this. …

So you have to to trial four times to have larger than 99% probability to have the first basket. If you trial $k=4$ times he of course may make more than one basket. … The probability distribution for the random number $Y$ of baskets made within $k$ trials then is binomial, as indicated by @Tim. …

Given a $2 \times 2$ contingency table with cell probabilities $$(\pi_{00}, \pi_{01},\pi_{10}, \pi_{11})$$ and marginal probabilities $$\pi_{+1}=\pi_{01}+\pi_{11}$$ and $$\pi_{1+}=\pi_{10}+\pi_{11},$$ …