Probability Distributions (Cambridge International A-Level Mathematics): Exam Questions

Two ordinary fair six-sided dice are rolled. The random variable $X$ is the number of sixes obtained.

(i) Find the probability of obtaining two sixes.

(ii) Complete the following probability distribution table for $X$.

Find the probability of obtaining at least one six.

The discrete random variable $X$ takes the values 0, 1, 2 and 3 only, and $\text{P} ( X = x ) = \frac{1}{4}$ for each of these values.

Draw up the probability distribution table for $X$.

Find:

(i) $\text{P} ( 1 \leq X \leq 2 )$

(ii) $\text{P} ( X < 3 )$

The discrete random variable $X$ takes the values 2 and 3 only, and $\text{P} ( X = x ) = k x$, where $k$ is a constant.

Use the fact that the sum of all probabilities equals 1 to show that $k = 0 . 2$.

Write down:

(i) $\text{P} ( 2 \leq X < 3 )$

(ii) $\text{P} ( X = 5 )$

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

Use the fact that the sum of all probabilities equals 1 to find the value of $p$.

Find:

(i) $\text{P} ( X \leq 4 )$

(ii) $\text{P} ( X > 7 )$

(iii) $\text{P} ( 2 \leq X \leq 6 )$

(iv) $\text{P} ( 3 < X < 7 )$

The discrete random variable $X$ takes the values 1, 2, 3 and 4 only. It is given that

$\text{P} ( X = x ) = k x$ when $x$ is odd

$\text{P} ( X = x ) = \frac{1}{2} k x$ when $x$ is even

where $k$ is a constant.

Show that $k = \frac{1}{7}$.

Draw up the probability distribution table for $X$, giving the probabilities as numerical fractions.

Show that $\text{P} ( X \leq 2 ) = \text{P} ( X = 4 )$.

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

Use the formula $\text{E} ( X ) = \sum x p$ to show that $\text{E} ( X ) = \frac{29}{12}$.

Use the formula $\text{E} ( X^{2} ) = \sum x^{2} p$ to show that $\text{E} ( X^{2} ) = \frac{95}{12}$.

Write down the formula that links $\text{Var} ( X )$, $\text{E} ( X )$ and $\text{E} ( X^{2} )$.

Hence show that $\text{Var} ( X ) = \frac{299}{144}$.

The probability distribution table for a discrete random variable $X$ is shown below.

Show that $p = \frac{3}{16}$.

Find $\text{E} ( X )$.

Show that $\text{E} ( X^{2} ) = \frac{103}{16}$.

Hence find $\text{Var} ( X )$.

The probability distribution table for a discrete random variable $X$ is shown below.

Show that $2 p + q = 0 . 7$.

Given that $\text{E} ( X ) = 3 . 3$, show that $7 p + 5 q = 2 . 9$.

Hence find the value of $p$ and the value of $q$.

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

Find:

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

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

(iii) $\text{P} ( 2 < X \leq 4 )$

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

Three biased coins are tossed. For each coin, the probability of obtaining a head is $\frac{2}{3}$. The random variable $X$ is the number of heads obtained.

Draw up the probability distribution table for $X$.

The random variable $X$ takes the values 1, 3, 5 and 7 only. The probability that $X$ takes the value $x$ is $k x$, where $k$ is a constant.

Find the value of $k$.

Find $\text{P} ( X > 3 )$.

The random variable $X$ takes the values −1, 0, 1, 2, 3 and 4 only. It is given that

$\text{P} ( X = - 1 ) = \text{P} ( X = 4 ) = 0 . 23$

$\text{P} ( X = 1 ) = \text{P} ( X = 3 ) = 0 . 13$

$\text{P} ( X = 0 ) = \text{P} ( X = 2 ) = k$

where $k$ is a constant.

Find the value of $k$.

Draw up the probability distribution table for $X$.

Find $\text{P} ( 0 \leq X < 3 )$.

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

Find the value of $p$.

Find $\text{E} ( X )$.

Find $\text{Var} ( X )$.

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

It is given that $\text{E} ( X ) = 0$.

Show that $p - 4 q = \frac{2}{15}$.

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

Hence find the values of $p$ and $q$.

Find $\text{Var} ( X )$.

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

Find the value of $p$.

Leonardo spins the spinner once. The random variable $X$ denotes the number on the sector where the spinner lands.

Find the probability that

(i) $X$ is no more than 1,

(ii) $X$ is at least 3.

Find the mean and the standard deviation of $X$.

Leonardo spins the spinner twice and adds the two scores together.

Find the probability that his total score is 5.

Three biased coins are tossed. For each coin, the probability of obtaining a head is $\frac{3}{5}$. The random variable $X$ is the number of heads obtained minus the number of tails obtained.

Draw up the probability distribution table for $X$.

The random variable $X$ takes the values −1, 1, 3 and 5 only.

Show that $\text{P} ( X = x ) = \frac{x^{2}}{30}$ cannot be the probability distribution of $X$.

It is given instead that $\text{P} ( X = x ) = \frac{x^{2}}{k}$, where $k$ is a constant.

Find the value of $k$.

Find $\text{P} ( X > 0 )$.

The random variable $X$ takes the values 0, 1, 3, 6, 10 and 15 only. It is given that

$\text{P} ( X = 0 ) = \text{P} ( X = 1 ) = 0 . 21$

$\text{P} ( X = 10 ) = \text{P} ( X = 15 ) = 0 . 11$

$\text{P} ( X = x ) = k x$ for $x = 3$ and $x = 6$

where $k$ is a constant.

Find the value of $k$.

Draw up the probability distribution table for $X$.

Find $\text{P} ( 3 < X \leq 14 )$.

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

Without calculating the value of $p$, explain why $\text{E} ( X ) = 2$.

Find the value of $p$.

Find $\text{Var} ( X )$.

Find \text{P}\left(X > \text{E}(X)\right).

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

Given that $\text{E} ( X ) = 2 . 45$, find the value of $p$ and the value of $q$.

Find $\text{Var} ( X )$.

Find \text{P}\left(X < \text{E}(X)\right).

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

Find the value of $p$.

Two independent observations of $X$ are taken. The random variable $Y$ is the sum of the two observations.

Complete the following probability distribution table for $Y$.

Find

(i) $\text{P} ( Y \neq 0 )$,

(ii) $\text{P} ( Y < 0 \text{or} Y \geq 2 )$.

Leonidas plays a game with a fair six-sided dice. He rolls the dice until either a 6 is obtained or he has rolled the dice four times. The random variable $X$ denotes the number of times 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.

Find the mean and the standard deviation of $X$.

Two biased coins are tossed and a fair spinner with three sectors numbered 1, 2 and 3 is spun. For each coin, the probability of obtaining a head is $\frac{1}{3}$. The random variable $X$ is the number of heads obtained multiplied by the number on the spinner.

Draw up the probability distribution table for $X$.

The random variable $X$ takes the values −3, −1, 1 and 3 only.

Show that

$\text{P} ( X = x ) = \frac{1}{3x^{2}}$ for $x = - 3 , - 1$

$\text{P} ( X = x ) = \frac{1}{3x^{3}}$ for $x = 1 , 3$

cannot be the probability distribution of $X$.

It is given instead that

$\text{P} ( X = x ) = \frac{k}{x^{2}}$ for $x = - 3 , - 1$

$\text{P} ( X = x ) = \frac{k}{x^{3}}$ for $x = 1 , 3$

where $k$ is a constant.

Find the exact value of $k$.

Find $\text{P} ( X < 2 )$.

The random variables $X$ and $Y$ are independent.

$X$ takes the values 1, 2, 3, 5, 8 and 11 only, each with probability $p$.

$Y$ takes the values 1, 3 and 6 only, and $\text{P} ( Y = y ) = \frac{q}{y}$, where $q$ is a constant.

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

Find $\text{P} ( X > Y )$.

Leofranc plays a game at a school fair. It costs 2 dollars to play. He chooses a "lucky number" from 1 to 6, then rolls a fair six-sided 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 dollars. If it first appears on the second, third or fourth roll, he receives 3 dollars, 2 dollars or 1 dollar respectively. If it has not appeared by the fourth roll, he receives nothing.

The random variable $W$ is Leofranc's profit, in dollars, from one game (the amount he receives minus the cost of playing). A negative profit means that he loses money.

Draw up the probability distribution table for $W$.

Find the probability that, in one game, Leofranc

(i) wins money,

(ii) loses money,

(iii) breaks even (neither wins nor loses money).

Find Leofranc's expected profit from one game.

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

Write down the value of $\text{E} ( X )$.

It is given that $\text{E} ( X^{2} ) = 3 . 4$.

Find $\text{Var} ( X )$.

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

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

It is given that $\text{E} ( X ) = 1 . 1$ and $\text{Var} ( X ) = 0 . 89$.

Find $\text{E} ( X^{2} )$.

Find the values of $a$, $b$ 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) $T$.

Determine which player is more likely to get a score greater than 2.

The random variable $X$ takes the values $p$, $2 p$, $3 p$, $4 p$ and $5 p$ only, where $p$ is a positive constant. It is given that $\text{P} ( X = x ) = \frac{x^{2}}{495}$.

Draw up the probability distribution table for $X$.

Find

(i) the mean, $μ$,

(ii) the standard deviation, $σ$,

of $X$.

Find $\text{P} ( μ - σ < X < μ + σ )$.