New Zealand
Level 8 - NCEA Level 3

# Problems with Discrete Random Variables and Probabilities

## Interactive practice questions

Two dice are rolled and the absolute value of the differences between the numbers appearing uppermost are recorded.

a

Complete the sample space.

 Die $2$2 1 2 3 4 5 6 Die $1$1 1 $0$0 $\editable{}$ $\editable{}$ $3$3 $\editable{}$ $\editable{}$ 2 $1$1 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ 3 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $2$2 $\editable{}$ 4 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ 5 $4$4 $\editable{}$ $2$2 $\editable{}$ $\editable{}$ $\editable{}$ 6 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
b

Let $X$X be defined as the absolute value of the difference between the two dice. Construct the probability distribution for $X$X using the table below.

Enter the values of $x$x from left to right in ascending order.

 $x$x $P$P$($($X=x$X=x$)$) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
c

Calculate $P$P$($($X<3$X<3$)$).

d

Calculate $P$P$($($X\le4$X4$|$|$X\ge2$X2$)$).

Easy
Approx 8 minutes

Two earrings are taken without replacement from a draw containing $3$3 black earrings and $5$5 brown earrings.

Let $X$X be the number of black earrings drawn.

An investment scheme advertises the following returns after $2$2 years based on historical probabilities.

A salesperson is starting work in a new region and analyses the probability of how many sales he is likely to make in the next month.

### Outcomes

#### S8-4

Investigate situations that involve elements of chance: A calculating probabilities of independent, combined, and conditional events B calculating and interpreting expected values and standard deviations of discrete random variables C applying distributions such as the Poisson, binomial, and normal

#### 91586

Apply probability distributions in solving problems