IB Maths AI HL 4.5 Probability basics Question Bank

Practise IB Mathematics SL/HL 4.5 by applying probability basics methods to exam-style questions.

[Maximum number: 5]

This question is about applying ideas from logarithms, calculus and probability to an unfamiliar mathematical theory called information theory.
Claude Shannon developed a mathematical theory called information theory to measure the information gained when random events occur. He defined the information, I, that is gained when an event with probability p occurs as

I=-\ln p

where $0<p \leq 1$ . For example, no information is gained ( I=0 ) when an event is certain to $\operatorname{occur}(p=1)$ .

(a)

A computer selects at random an integer x from 1 to 10, inclusive. Each outcome is equally likely.

Alessia is trying to determine the value of x and asks if x is odd.

[ 5 ]

(i)

Write down the probability that x is odd.

[ 1 ]

$\frac{5}{10} \quad\left(=\frac{1}{2}\right)$

(ii)

Alessia is told that x is odd. Find how much information Alessia gains.

The computer then selects at random an integer y from 1 to 10 , inclusive. Each outcome is equally likely.

Daniel is trying to determine the value of y and asks if y is 7 . He is told that it is not 7 .

[ 2 ]

attempt to substitute p= their (b)(i) into $I=-\ln p$ $=0.693 \quad\left(0.693147 \ldots,-\ln \left(\frac{1}{2}\right)\right)$

(iii)

Find how much information Daniel gains.

If a random variable has n possible outcomes with probabilities $p_{1}, p_{2} \ldots p_{n}$ , then the expected information gained, E(I), is defined as

\mathrm{E}(I)=\sum_{r=1}^{n}-p_{r} \ln p_{r}

[ 2 ]

$\frac{9}{10}$ $=0.105 \quad\left(0.105360 \ldots,-\ln \left(\frac{9}{10}\right)\right)$

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SL 4.5—Probability basics question 1

Question (a)

Question (i)

Question (ii)

Question (iii)