Statistics and probability practice questions (IB Maths AA)

Sample questions

The one free IB Maths AA question on this topic, from the free IB Maths AA SL Paper 1 Practice Exam. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 16 marks · Conditional probability & independence
   The probability that it rains on a given morning is $\frac{1}{4}$. If it rains, the probability that Mia cycles to school is $\frac{1}{5}$; if it does not rain, the probability that she cycles is $\frac{3}{5}$. Let $R$ be the event "it rains" and $C$ the event "Mia cycles". A tree diagram may help.
   (a)

   Find $P(C)$, the probability that Mia cycles to school.

   [2]
   (b)

   Given that Mia cycles to school, find the probability that it rained.

   [2]
   (c)

   Determine, with a reason, whether the events $R$ and $C$ are independent.

   [2]
   Show answerFrom the free IB Maths AA SL Paper 1 Practice Exam →

Key terms

The definitions and formulas IB Maths AA expects for statistics and probability, typeset as they appear on the paper.

Conditional probability

The probability of $A$ given that $B$ has happened. Rearranged, it gives the multiplication rule $P(A \cap B) = P(A|B)\,P(B)$ that drives tree diagrams.

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$

Independent events

Two events are independent when knowing one has happened does not change the probability of the other. Show independence by checking the product rule, not by asserting it.

$$P(A \cap B) = P(A)\,P(B) \iff P(A \mid B) = P(A)$$

Binomial distribution

The number of successes in $n$ independent trials with constant probability $p$. State the distribution with its parameters before calculating; on Paper 2 the GDC gives the probabilities.

$$X \sim B(n, p), \qquad P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}, \qquad E(X) = np, \quad Var(X) = np(1-p)$$

Normal distribution

The bell-shaped continuous distribution with mean $\mu$ and standard deviation $\sigma$. Standardising converts any normal value into a $z$-score, which inverse-normal questions use to find an unknown $\mu$ or $\sigma$.

$$X \sim N(\mu, \sigma^2), \qquad z = \frac{x - \mu}{\sigma}$$

Expected value of a discrete random variable

The probability-weighted mean of the outcomes. A game is fair when the expected gain is zero; a probability distribution is valid only when its probabilities sum to 1.

$$E(X) = \sum x\,P(X = x)$$

Measures of spread and outliers

Range, interquartile range, variance and standard deviation. A value is an outlier when it lies more than $1.5 \times IQR$ beyond the nearest quartile.

$$\text{outlier if } x < Q_1 - 1.5\,IQR \text{ or } x > Q_3 + 1.5\,IQR$$

Linear regression and correlation

Pearson's $r$ measures the strength and direction of a linear relationship. The regression line of $y$ on $x$ passes through $(\bar{x}, \bar{y})$ and should be used to predict $y$ only within the range of the data.

Frequently asked questions

Other IB Maths AA topics

• Number and algebra
• Functions
• Geometry and trigonometry
• Calculus

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