The principle of inclusion and exclusion is a counting principle used to calculate the probability of the union of multiple events. It's based on the idea that the union's probability should add the individual probabilities and subtract the probabilities of intersections to avoid double-counting.

When we say that a distribution is skewed, we mean that the distribution is not symmetric about its mean. In a skewed distribution, the data points are not evenly distributed around the mean, with more data on one side of the mean than the other.

If the p-value in a Chi-square test is smaller than the significance level, we reject the null hypothesis in favor of the alternative hypothesis. This suggests that there is a significant association between the variables.

Multicollinearity refers to a situation where two or more predictor variables in a multiple regression model are highly correlated. This high correlation can result in unstable coefficient estimates, making them less reliable and harder to interpret.

When data points are concentrated on the left and the tail is on the right, the distribution is said to be positively skewed or right-skewed. This is because the tail of the distribution points towards the positive end of the axis.

Residual analysis is important because it helps us to validate the assumptions of the regression model, such as linearity, independence, normality, and equal variance (homoscedasticity). This is crucial for the reliability and validity of the regression model.

The Total Probability Rule provides a way to compute the probability of an event from the probabilities of that event occurring within disjoint subsets of the sample space. It essentially allows you to break down the probability of complex events into simpler or more basic component events.

In a Chi-square test for goodness of fit, the degrees of freedom are calculated as the number of categories minus one. This reflects the number of values in the final calculation that are free to vary.

The choice of bin size in a histogram can greatly affect the resulting visualization. If the bins are too large, important features of the data may be obscured. If the bins are too small, the histogram may appear too 'noisy' and it may be difficult to interpret underlying patterns. Thus, the choice of bin size can indeed change the perceived shape of the histogram.

Heteroscedasticity can be resolved by transforming the dependent variable, typically using a logarithmic transformation. This often stabilizes the variance of the residuals across different levels of the predictors.