Revise Discrete random variables and expected value for IB Mathematics: Applications and Interpretation (4.5)

A discrete random variable assigns a number to each outcome of a random experiment, and takes only separate, countable values - for example the number of heads in four coin flips, which can be $0,1,2,3$ or $4$ but never $2.5$. We write the variable with a capital letter, $X$, and a particular value with a lowercase $x$. This is the difference from a continuous variable such as height, which can take any value in a range.

A probability distribution lists every value the variable can take alongside its probability $P(X=x)$, almost always as a table. The table is the object every 4.5 question works from, so building it correctly - and completely - is the first skill to master.

The condition $\sum P(X=x)=1$ is more than a check: when a distribution contains an unknown probability - often written in terms of a constant $k$ - you find it by setting the total equal to 1 and solving.

The expected value of a discrete random variable is its theoretical long-run average - the mean you would approach if you repeated the experiment a very large number of times. It is a probability-weighted average: each value is weighted by how likely it is, so common outcomes count for more than rare ones. It is also called the mean and written $E(X)$ or $\mu$.

Expected value: $E(X) = \mu = \sum x\,P(X=x)$.

To compute it, multiply each value $x$ by its probability $P(X=x)$ and add the products. Two ideas trip students up. First, this is a weighted average - averaging the values without their probabilities is only correct when every outcome is equally likely. Second, the expected value need not be a value the variable can actually take: a fair die has $E(X)=3.5$, and you never round that to a real face.

Expected value is the tool for judging games of chance. A game is fair when the player's expected gain is $0$ - on average the player neither wins nor loses. The expected gain is the expected winnings minus the cost to play: $E(\text{gain}) = E(W) - \text{stake}$. If the expected gain is positive the game favours the player; if negative it favours the organiser (which is how casinos make money). The single most common error here is comparing the expected winnings with zero and forgetting to subtract the stake.

On Paper 2 the marks are analytic: each is tied to a specific line of working - a method mark (M) or an accuracy mark (A) - and an accuracy mark depends on the method mark it follows. Follow-through (FT) means an earlier slip need not cost the marks that depend on it, provided the later step is carried out correctly on your own figures. The engine also ignores subsequent working (ISW) once a correct answer appears and accepts any equivalent form or correctly-rounded value - but only if the method is on the page. Study how each mark below is earned by a specific line.

The distribution table and the weighted sum you have built here are the template for every named distribution in the course. The binomial and, later, other models are just probability distributions with a formula for $P(X=x)$ instead of a hand-filled table, and their means are the same expected value $E(X)=\sum x\,P(X=x)$ specialised to that formula. The fair-game idea generalises to expected profit in any decision under uncertainty - insurance premiums, expected returns, risk. Master the two moves in this lesson - make the probabilities sum to 1, then form the probability-weighted average - and the rest of the probability course is variations on a calculation you already know how to lay out.