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Counting: permutations and combinations
Factorials, ordered and unordered selections.
n! counts the orderings of n things. Choosing k of them in order is n!/(n − k)!; ignoring the order divides by k! again, giving the binomial coefficient “n choose k”. These numbers are Pascal's triangle and the coefficients of (a + b)ⁿ.
Mfano wenye matokeo · 5 choose 2
Hatua kwa hatua
- \binom{5}{2} = \frac{5!}{2!\,(5-2)!}
Unordered selections: n! / (k! (n−k)!).
- = \frac{120}{2 \times 6} = 10
Lafunua jibu
Sasa Chagua tatizo, au aina ya barua au uchore.
Ishara zinazotumiwa hapa
Vaa alama yoyote ya ufafanuzi kamili, picha, na maana ya kila herufi.
Jinsi ya Kufaulu: Counting: permutations and combinations
- Unordered selections: n! / (k! (n−k)!).
Maswali ambayo watu huuliza
Mean or median, which should I use?
Median when the data have outliers or a long tail (incomes, house prices); mean when the data are roughly symmetric and you want every value to count. Report both if they disagree. The gap is itself information.
What does a p-value actually say?
The probability of seeing data at least this extreme if the null hypothesis were true. It is not the probability that the null hypothesis is true.
Why divide by n − 1 for the sample variance?
The sample mean sits closer to the sample than the true mean does, so squared deviations from it are slightly too small on average; dividing by n − 1 instead of n corrects the bias.
Mengi zaidi katika Statistics & Probability
Sampling and dataDescribing data with graphsMean, median and modeProbabilityDiscrete random variablesContinuous random variablesThe normal distributionThe central limit theoremConfidence intervalsHypothesis testingComparing two samplesChi-square testsLinear regression and correlationANOVA and the F distribution