Statistics & Probability (AHL)

The random variable $X$ has $\mathrm{Var}(X) = 2$. Find $\mathrm{Var}(2X + 1)$.

A discrete random variable $X$ takes the values $1, 2, 3, 4$ with probabilities $\frac{1}{8}, \frac{1}{4}, \frac{3}{8}, \frac{1}{4}$ respectively. Find $\mathrm{E}(X)$.

A bag contains $3$ red and $2$ blue counters. Two counters are drawn at random without replacement. Find the probability that both are red.

A random variable takes the value $0$ with probability $\frac{3}{4}$ and $2$ with probability $\frac{1}{4}$. Find its variance.

Machine $A$ produces $\frac{3}{5}$ of a factory's output and machine $B$ the rest. $\frac{1}{20}$ of $A$'s items and $\frac{1}{10}$ of $B$'s items are defective. An item is chosen at random and found to be defective. Find the probability it was made by machine $A$.

One of two bags is chosen at random. Bag 1 gives a red ball with probability $\frac{2}{10}$ and bag 2 with probability $\frac{3}{10}$. A red ball is drawn. Find the probability it came from bag 1.

The random variable $X$ has $\mathrm{Var}(X) = 5$. Find $\mathrm{Var}(2X + 3)$.

A discrete random variable $X$ takes the values $1, 2, 3, 4$ with probabilities $\frac{1}{10}, \frac{1}{5}, \frac{2}{5}, \frac{3}{10}$ respectively. Find $\mathrm{E}(X)$.

A bag contains $5$ red and $5$ blue counters. Two counters are drawn at random without replacement. Find the probability that both are red.

A random variable takes the value $0$ with probability $\frac{9}{10}$ and $5$ with probability $\frac{1}{10}$. Find its variance.

Machine $A$ produces $\frac{7}{10}$ of a factory's output and machine $B$ the rest. $\frac{1}{50}$ of $A$'s items and $\frac{3}{50}$ of $B$'s items are defective. An item is chosen at random and found to be defective. Find the probability it was made by machine $A$.

One of two bags is chosen at random. Bag 1 gives a red ball with probability $\frac{2}{12}$ and bag 2 with probability $\frac{6}{12}$. A red ball is drawn. Find the probability it came from bag 1.