Higher Level Mathematics

Ibrahim Wazir, Tim Garry, Peter Ashbourne

Chapter 17

Probability Distributions - all with Video Answers

Educators

Section 1

Random variables

Problem 1

Classify each of the following as discrete or continuous random variables.
a) The number of words spelled correctly by a student on a spelling test.
b) The amount of water flowing through the Niagara Falls per year.
c) The length of time a student is late to class.
d) The number of bacteria per cc of drinking water in Geneva.
e) The amount of CO produced per litre of unleaded gas.
f) The amount of a flu vaccine in a syringe.
g) The heart rate of a lab mouse.
h) The barometric pressure at Mount Everest.
i) The distance travelled by a taxi driver per day.
j) Total score of football teams in national leagues.
k) Height of ocean tides on the shores of Portugal.
1) Tensile breaking strength (in newtons per square metre) of a 5 cm diameter steel cable.
m) Number of overdue books in a public library.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 2

A random variable Y has this probability distribution:
$$\begin{array}{|l|c|c|c|c|c|c|}
\hline y & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline P(y) & 0.1 & 0.3 & & 0.1 & 0.05 & 0.05 \\
\hline
\end{array}$$
a) Find $P(2)$
b) Construct a probability histogram for this distribution.
c) Find $\mu$ and $\sigma$
d) Locate the interval $\mu \pm \sigma$ as well as $\mu \pm 2 \sigma$ on the histogram.
e) We create another random variable $Z=b+1 .$ Find $\mu$ and $\sigma$ of $Z$
f) Compare your results for $c$ ) and e) and generalize for $Z=Y+b,$ where $b$ is a constant.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 3

A discrete random variable $X$ can assume five possible values: 12,13,15,18 and
20. Its probability distribution is shown below.
$$\begin{array}{|l|c|c|c|c|c|} \hline x & 12 & 13 & 15 & 18 & 20 \\ \hline P(x) & 0.14 & 0.11 & & 0.26 & 0.23 \\\hline \end{array}$$
a) What is $P(15) ?$
b) What is the probability that $x$ equals 12 or $20 ?$
c) What is $P(x \leqslant 18) ?$
d) Find $E(X)$
e) Find $V(X)$
f) Let $Y=0.5 X-4 .$ Find $E(Y)$ and $V(n)$
g) Compare your results in $\mathrm{d}$ ), $\mathrm{e}$ ) and $\mathrm{f}$ ) and generalize for $Y=a X+b,$ where $a$ and $b$ are constants.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 4

Medical research has shown that a certain type of chemotherapy is successful $70 \%$ of the time when used to treat skin cancer. In a study to check the validity of such a claim, researchers chose different treatment centres and chose five of their patients at random. Here is the probability distribution of the number of successful treatments for groups of five:
$$\begin{array}{|l|c|c|c|c|c|c|}
\hline x & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline P(x) & 0.002 & 0.029 & 0.132 & 0.309 & 0.360 & 0.168 \\
\hline
\end{array}$$
a) Find the probability that at least two patients would benefit from the treatment.
b) Find the probability that the majority of the group does not benefit from the treatment.
c) Find $E(X)$ and interpret the result.
d) Show that $\sigma(x)=1.02$
e) Graph $P(x)$. Locate $\mu, \mu \pm \sigma$ and $\mu \pm 2 \sigma$ on the graph. Use the empirical rule
to approximate the probability that $x$ falls in this interval. Compare this with the actual probability.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 5

The probability function of a discrete random variable $X$ is given by
$$P(X=x)=\frac{k x}{2}, \text { for } x=12,14,16,18$$
Set up the table showing the probability distribution and find the value of $k$.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 6

$X$ has probability distribution as shown in the table.
$$\begin{array}{|l|c|c|c|c|c|}
\hline x & 5 & 10 & 15 & 20 & 25 \\
\hline P(x) & \frac{3}{20} & \frac{7}{30} & k & \frac{3}{10} & \frac{13}{60} \\
\hline
\end{array}$$
a) Find the value of $k$
b) Find $P(x>10)$
c) Find $P(5<x \leqslant 20)$
d) Find the expected value and the standard deviation.
e) Let $Y=\frac{1}{5} X-1 .$ Find $E(Y)$ and $V(Y)$

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 7

The discrete random variable Y has a probability density function
$$P(Y=y)=k\left(16-y^{2}\right), \text { for } y=0,1,2,3,4$$
a) Find the value of the constant $k$
b) Draw a histogram to illustrate the distribution.
c) Find $P(1 \leqslant y \leqslant 3)$
d) Find the mean and variance.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 8

The probability distribution of students categorized by age that visit a certain movie house on weekends is given on the right. The probabilities for $18-$ and 19 -year-olds are missing. We know that
$$P(x=18)=2 P(x=19)$$.
(GRAPHS CANNOT COPY)
a) Complete the histogram and describe the distribution.
b) Find the expected value and the variance.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 9

In a small town, a computer store sells laptops to the local residents. However, due to low demand, they like to keep their stock at a manageable level. The data they have indicate that the weekly demand for the laptops they sell follows the distribution given in the table below.
(TABLE CANNOT COPY)
a) Find the mean and standard deviation of this distribution.
b) Use the empirical rule to find the approximate number of laptops that is sold about $95 \%$ of the time.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 10

The discrete random variable $X$ has probability function given by
$$P(x)=\left\{\begin{array}{cc}
\left(\frac{1}{4}\right)^{x-1} & x=2,3,4,5,6 \\
k & x=7 \\
0 & \text { otherwise }
\end{array}\right.$$
where $k$ is a constant. Determine the value of $k$ and the expected value of $X .$

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 11

The following is a probability distribution for a random variable $Y$.
$$\begin{array}{|l|c|c|c|c|}
\hline y & 0 & 1 & 2 & 3 \\
\hline P(Y=y) & 0.1 & 0.11 & k & (k-1)^{2} \\
\hline
\end{array}$$
a) Find the value of $k$
b) Find the expected value.

Willis James

Numerade Educator

Problem 12

A closed box contains eight red balls and four white ones. A ball is taken out at random, its colour noted, and then returned. This is done three times. Let $x$ represent the number of red balls drawn.
a) Set up a table to show the probability distribution of $X$.
b) What is the expected number of red balls in this experiment?

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 13

A discrete random variable $Y$ has the following probability distribution function:
$$
P(Y=y)=k(4-y), \text { for } y=0,1,2,3 \text { and } 4
$$
a) Find the value of $k$
b) Find $P(1 \leqslant y<3)$

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 14

Airlines sometimes overbook flights. Suppose for a 50 -seat plane, 55 tickets were sold. Let $X$ be the number of ticketed passengers that show up for the flight. From records, the airline has the following pmf for this flight.
$$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|}
\hline x & 45 & 46 & 47 & 48 & 49 & 50 & 51 & 52 & 53 & 54 & 55 \\
\hline P(x) & 0.05 & 0.08 & 0.12 & 0.15 & 0.25 & 0.20 & 0.05 & 0.04 & 0.03 & 0.02 & 0.01 \\
\hline
\end{array}$$
a) Construct a cuf table for this distribution.
b) What is the probability that the flight will accommodate all ticket holders that show up?
c) What is the probability that not all ticket holders will have a seat on the flight?
d) Calculate the expected number of passengers who will show up.
e) Calculate the standard deviation of the passengers who will show up.
f) Calculate the probability that the number of passengers showing up will be within one standard deviation of the expected number.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 15

A small internet provider has 6 telephone service lines operating 24 -hours daily. Defining $X$ as the number of lines in use at any specific 10 -minute period of the day, the pmf of $X$ is given in the following table.
$$\begin{array}{|l|c|c|c|c|c|c|c|}
\hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline P(x) & 0.08 & 0.15 & 0.22 & 0.27 & 0.20 & 0.05 & 0.03 \\
\hline
\end{array}$$
a) Construct a cdf table.
b) Calculate the probability that at most three lines are in use.
c) Calculate the probability that a customer calling for service will have a free line.
d) Calculate the expected number of lines in use.
e) Calculate the standard deviation of the number of lines in use.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 16

Some flashlights use one AA-type battery. The voltage in any new battery is considered acceptable if it is at least 1.3 volts. $90 \%$ of the AA batteries from a specific supplier have an acceptable voltage. Batteries are usually tested till an acceptable one is found. Then it is installed in the flashlight. Let $X$ be the number of batteries that must be tested.
a) What is $P(1),$ i.e. $P(x=1) ?$
b) What is $P(2) ?$
c) What is $P(3) ?$
d) To have $x=5,$ what must be true of the fourth battery tested? of the fifth one?
e) Use your observations above to obtain a general model for $P(x)$.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 17

Repeat question $16,$ but now consider the flashlight as needing two batteries.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 18

A biased die with four faces is used in a game. A player pays 10 counters to roll the die. The table below shows the possible scores on the die, the probability of each score and the number of counters the player receives in return for each score.
$$\begin{array}{|l|c|c|c|c|}
\hline \text { Score } & 1 & 2 & 3 & 4 \\
\hline \text { Probability } & \frac{1}{2} & \frac{1}{5} & \frac{1}{5} & \frac{1}{10} \\
\hline \text { Number of counters player receives } & 4 & 5 & 15 & n \\
\hline
\end{array}$$
Find the value of $n$ in order for the player to get an expected return of 9 counters per roll.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 19

Two children, Alan and Belle, each throw two fair cubical dice simultaneously. The score for each child is the sum of the two numbers shown on their respective dice.
a) (i) Calculate the probability that Alan obtains a score of 9
(ii) Calculate the probability that Alan and Belle both obtain a score of $9 .$
b) (i) Calculate the probability that Alan and Belle obtain the same score.
(ii) Deduce the probability that Alan's score exceeds Belle's score.
c) Let $X$ denote the largest number shown on the four dice.
(i) Show that for $P(X \leqslant x)=\left(\frac{x}{6}\right)^{4},$ for $x=1,2, \ldots, 6$
(ii) Copy and complete the following probability distribution table.
$$\begin{array}{|l|c|c|c|c|c|c|}
\hline x & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline P(X=x) & \frac{1}{1296} & \frac{15}{1296} & & & & \frac{671}{1296} \\
\hline
\end{array}$$
(iii) Calculate $E(X)$

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 20

$$\begin{array}{l}
\text { Consider the } 10 \text { data items } x_{1}, x_{2}, \ldots, x_{10} \text { . Given that } \sum_{i=1}^{10} x^{2},=1341 \text { and the } \\
\text { standard deviation is } 6.9, \text { find the value of } \bar{x} .
\end{array}$$.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator