Probability Distributions (Cambridge (CIE) A Level Maths: Probability & Statistics 1): Exam Questions

Three biased coins are tossed.

 Write down all the possible outcomes when the three coins are tossed.

A random variable, X , is defined as the number of heads when the three coins are tossed.

Given that for each coin the probability of getting heads is  $\frac{2}{3}$,

complete the following probability distribution table for X:

The random variable X  has the probability function

         $\mathrm{P}\left(X=x\right)=\{\begin{array}{l}kxx=1,3,5,7\\ 0\mathrm{otherwise}\end{array}$

Find the value of $k$.

Find $\mathrm{P}\left(X>3\right).$

State, with a reason, whether or not $X$ is a discrete random variable.

The random variable X has the probability function

$\mathrm{P}\left(X=x\right)=\{\begin{array}{l}0.23x=-1,4\\ kx=0,2\\ 0.13x=1,3\\ 0\mathrm{otherwise}\end{array}$

Find the value of $k$.

Construct a table giving the probability distribution of X.

Find $\mathrm{P}\left(0\le X<3\right).$

A discrete random variable  X has the probability distribution shown in the following table:

Find:

(i) $\mathrm{P}(X<4)$

(ii) $\mathrm{P}(X>1)$

(iii) $\mathrm{P}(2<X\le 4)$

(iv) $\mathrm{P}(0<X<4)$

The discrete random variable $X$ has the probability distribution shown in the following table: 

 Find the value of $p$.

Find $\mathrm{E}\left(X\right)$.

Find $V\mathrm{ar}\left(X\right)$.

The discrete random variable $X$ has the probability distribution shown in the following table:

 2

It is given that $E\left(X\right)=0.$

Show that $p-4q=\frac{2}{15}$.

Write down a second equation involving $p$ and $q$.

Hence find the values of $p$ and $q$.

Find $\mathrm{Var}\left(X\right)$.

Leonardo has constructed a biased spinner with six sectors labelled 0,1, 1, 2, 3 and 5.  The probability of the spinner landing on each of the six sectors is shown in the following table:

Find the value of $p$.

Leonardo is playing a game with his biased spinner.  The score for the game, $X$, is the number which the spinner lands on after being spun.

Find the probability that Leonardo’s score is

(i) no more than 1

(ii) at least 3.

(i) Find the expected value for Leonardo’s score in a game.

(ii) Find the standard deviation of Leonardo’s scores.

Leonardo plays the game twice and adds the two scores together. Find the probability that Leonardo has a total score of 5.

Three biased coins are tossed.

 Write down all the possible outcomes when the three coins are tossed.

A random variable, X, is defined as the number of heads when the three coins are tossed minus the number of tails.

Given that for each coin the probability of getting heads is $\frac{3}{5}$,

complete the following probability distribution table for X:

A student claims that a random variable X  has a probability distribution defined by the following probability mass function:

$P\left(X=x\right)=\{\begin{array}{l}\frac{x^2}{30}x=-1,1,3,5\\ 0\mathrm{otherwise}\end{array}$

Explain how you know that the student’s function does not describe a probability distribution.

Given that the correct probability mass function is of the form

$P\left(X=x\right)=\{\begin{array}{l}\frac{x^2}{k}x=-1,1,3,5\\ 0\mathrm{otherwise}\end{array}$

where $k$ is a constant,

find the exact value of $k$.

Find $\mathrm{P}(X>0).$

State, with a reason, whether or not $X$ is a discrete random variable.

The random variable  X has the probability function

$P\left(X=x\right)=\{\begin{array}{l}0.21x=0,1\\ kxx=3,6\\ 0.11x=10,15\\ 0\mathrm{otherwise}\end{array}$

Find the value of $k$.

Construct a table giving the probability distribution of X.

Find $\mathrm{P}(3<X\le 14)$

A discrete random variable  has the probability distribution shown in the following table:

Find the value of $p$.

X  is sampled twice such that the results of the two experiments are independent of each other, and the outcomes of the two experiments are recorded.  A new random variable,Y, is defined as the sum of the two outcomes.

Complete the following probability distribution table for Y:

Find:

(i) $\mathrm{P}(Y\ne 0)$

(ii) $\mathrm{P}(Y>1)$

(iii) $\mathrm{P}(-2<Y<2)$

(iv) $\mathrm{P}(Y<0orY\ge 2)$

Leonidas is playing a game with a fair six-sided dice on which the faces are numbered 1 to 6.  He rolls the dice until either a ‘6’ appears or he has rolled the dice four times. The random variable X is defined as the number of times that the dice is rolled.

Draw up the probability distribution table for X.

Find the probability that the dice is rolled

(i) at most 3 times.

(ii) at least 3 times.

(i) Find the expected number of times that Leonidas will roll the dice.

(ii) Find the standard deviation of the number of times that Leonidas will roll the dice.

The discrete random variable $X$ has the probability distribution shown in the following table: 

Without working out the value of $p$, explain why $\mathrm{E}\left(X\right)=2.$

find the value of $p$.

Find $\mathrm{Var}\left(\mathrm{X}\right)$.

The outcome of a random variable $Y$ is double the outcome of $X$. Complete the probability distribution for $Y$ below: 

Find $P(Y>X)$.

The discrete random variable $X$ has the probability distribution shown in the following table: 

 It is given that $\mathrm{E}\left(X\right)$ $=2.45.$ 

Find the values of $p$ and $q$.

Find $\mathrm{Var}\left(X\right)$

Find $\mathrm{P}(X<E(X\left)\right).$

Two biased coins are tossed and a fair spinner with three sectors numbered 1 to 3 is spun.

Write down all the possible outcomes when the two coins are tossed and the spinner is spun.

A random variable, $X$, is defined as the number of heads when the two coins are tossed multiplied by the number the spinner lands on when it is spun.

For each coin the probability of getting heads is  $\frac{1}{3}$.

Complete the following probability distribution table for $X$:

A student claims that a random variable $X$ has a probability distribution defined by the following probability mass function:

$P\left(X=x\right)=\{\begin{array}{l}\frac{1}{3x^2}x=-3,-1\\ \frac{1}{3x^3}x=1,3\\ 0\mathrm{otherwise}\end{array}$

Explain how you know that the student’s function does not describe a probability distribution.

Given that the correct probability mass function is of the form

 $P\left(X=x\right)=\{\begin{array}{l}\frac{k}{x^2}x=-3,-1\\ \frac{k}{x^3}x=1,3\\ 0\mathrm{otherwise}\end{array}$

where $k$ is a constant,

 Find the exact value of $k$.

Find $\mathrm{P}\left(X<2\right)$.

State, with a reason, whether or not $X$ is a discrete random variables.

The random variable $X$ has the probability function

$P\left(X=x\right)=\frac{x^2}{495},x=p,2p,3p,4p,5p$

where  $p>0$is a constant.

 Construct a table giving the probability distribution of X.

Find:

   (i)        the mean $\mu$,

   (ii)       the standard deviation, $\sigma$,

of $X$.

Find $\mathrm{P}\left(\mu -\sigma <X<\mu +\sigma \right)$

The independent random variables X  and Y  have probability distributions

                   $\mathrm{P}(X=x)=p,x=1,2,3,5,8,11$

                  $\mathrm{P}(Y=y)=\frac{q}{y},y=1,3,6$

where $p$ and $q$ are constants.

 Find  $\mathrm{P}(X>Y)$.

Leofranc is playing a gambling game with a fair six-sided dice on which the faces are numbered 1 to 6.  He must pay £2 to play the game.  He then chooses a ‘lucky number’ between 1 and 6, and rolls the dice until either his lucky number appears or he has rolled the dice four times.  If his lucky number appears on the first roll, he receives £5 back.  If his lucky number appears on the second, third or fourth rolls, he receives £3, £2 or £1 back respectively.  If his lucky number has not appeared by the fourth roll, then the game is over and he receives nothing back.

 The random variable $W$ is defined to be Leofranc’s profit (i.e., the amount of money he receives back minus the cost of playing the game) when he plays the game one time.  Note that a negative profit indicates that Leofranc has lost money on the game.

Draw up the probability distribution table for $W$.

Find the probability that when playing the game one time Leofranc

(i) wins money

(ii) loses money

(iii) breaks even (i.e., does not lose money).

Find the expected profit after one game.

The discrete random variable  has the probability distribution shown in the following table: 

Write down the value of $\mathrm{E}\left(X\right)$.

It is given that $\mathrm{E}\left(X^2\right)=3.4$. 

Find $\mathrm{Var}\left(X\right)$.

Find the values for $p$ and $q$.

A spinner has three sectors labelled 0, 1 and 2. Let $X$ be the random variable denoting the number the spinner lands on when spun. The probability distribution table for $X$ is shown below: 

 It is given that $\mathrm{E}\left(X\right)$$=1.1$ and $\mathrm{Var}\left(X\right)=0.89$. 

Write down the value of $\mathrm{E}\left(X^2\right)$.

Find the values of $a,b\mathrm{and}c$.

Susie spins the spinner twice and adds together the two numbers to calculate her score, $S$. Tommy spins the spinner once and doubles the number to calculate his score, $T$. Each spin of the spinner is independent of all other spins. 

Draw up the probability distribution table for:

(i) $S$,

(ii) $T$.

Which player is most likely to get a score that is bigger than 2?