IB Mathematics Analysis and Approaches Topic SL…

A discrete random variable $W$ models the number of customers arriving at a counter in a 10-minute period, with the probability distribution:

(a) Find the value of $p$ and hence complete the probability distribution.

(b) Find $E(W)$ (customers per 10 minutes).

(a) Find the value of $p$ and hence complete the probability distribution.

A game uses a fair six-sided die. Let $W$ denote the amount won, in dollars, in a single play of the game:

if a $1$ is rolled, the player loses $4;

if a $2$ or a $3$ is rolled, the player loses $1;

if a $4$ or a $5$ is rolled, the player wins $2;

if a $6$ is rolled, the player wins $k.

(a) Show that $E(W) = \dfrac{k - 2}{6}$ .

(b) Find the value of $k$ for which the game is fair.

An online game offers mystery boxes, where each box has a 10% chance of containing a bonus code. Let $N$ be the random variable representing the number of boxes opened to obtain two bonus codes.

(a) Show that $P(N=2)=0.01$ and find $P(N=3)$ .

(b) Find $E(N)$ and the probability that at most 8 boxes are needed.

(c) If the bonus code is not redeemed, its value in week $k$ is $75 \cdot 3^{k-1}$ credits. Find the smallest $k$ such that the expected bonus-code value for a box opened in week $k$ exceeds 50 credits.

(d) Find the mode(s) of the probability distribution.

A discrete random variable $X$ models the number of messages received by a help desk in a 5-minute period, with the probability distribution:

$x$	0	2	3
$P(X=x)$	0.25	$p$	$0.4p$

(b) Find $E(X)$ (messages per 5 minutes).

[Maximum mark: 16] A cinema promotion sells mystery envelopes, where each envelope has a 30% chance of containing a gift voucher. Let $W$ be the random variable representing the number of envelopes purchased to obtain two gift vouchers.

(a) Show that $P(W=2)=0.09$ and find $P(W=3)$ .

(b) Find $E(W)$ and the probability that at most 6 envelopes are needed.

(c) If the voucher is not claimed, its value in week $k$ is $10 \cdot 2^{k-1}$ dollars. Find the smallest $k$ such that the expected prize value for an envelope purchased in week $k$ exceeds 20 dollars.

(d) Find the mode(s) of the probability distribution.

IB Mathematics Analysis and Approaches SL 4.7—discrete Random Variables Questionbank

IB Mathematics Analysis and Approaches SL 4.7—discrete Random Variables Questionbank

IB Mathematics Analysis and Approaches SL 4.7—discrete Random Variables Questionbank

IB Mathematics Analysis and Approaches SL 4.7—discrete Random Variables Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 4.7—discrete Random Variables Questionbank