Probability Distributions (DP IB Applications & Interpretation (AI): HL): Exam Questions

Three biased coins are tossed.

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

For each coin the probability of getting heads is $\frac{2}{3}$. A random variable, $X$, is defined as the number of heads when the three coins are tossed.

Complete the following probability distribution table for $X$.

Hence, by inserting the relevant probabilities, represent the probability distribution for $X$ as a piecewise function in the form

$\mathrm{P}(X=x)=\{\begin{array}{cc}& x=0\\ & x=1\\ & x=2\\ & x=3\\ 0,& \text{otherwise}\end{array}$

Represent the probability distribution for $X$ as a bar chart.

The random variable $X$ has the probability function

$\mathrm{P}(X=x)=\{\begin{array}{cc}kx,& x=1,3,5,7\\ 0,& \text{otherwise}\end{array}$

Find the value of $k$.

Find $P(X>3)$.

State whether $X$ is discrete or continuous.

The random variable $X$ has the probability function

$\mathrm{P}(X=x)=\{\begin{array}{cc}0.23,& x=-1,4\\ k,& x=0,2\\ 0.13,& x=1,3\\ 0,& \text{otherwise}\end{array}$

Find the value of $k$.

Complete the following table to show the probability distribution of $X$.

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

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.

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

Find the probability that $X$ is

(i) no more than 1

(ii) at least 3.

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

where $k$ is a positive constant.

Show that $k^2+2k-0.69=0$.

Hence find the value of $k$, giving a reason for your answer.

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

A spinner is divided into five sections, A, B, C, D and E.

The probability of the spinner landing on each section is given by the following table.

A person who spins the spinner scores points depending on which section the spinner lands on. These points are shown below.

Given that the game is fair, find the value of $k$.

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

It is given that $\mathrm{E}\left(X\right)=2.3$.

Show that $a$ and $b$ satisfy the following two equations.

$a+b=0.75$

$2a+3b=1.85$

Hence find the value of $a$ and the value of $b$.

Find $P(1\le X<4)$.

Frank plays a game involving a biased six-sided die.

The faces of the die are numbered 1 to 6.

The score of the game, $X$, is the number which lands face up after the die is rolled.

The following table shows the probability distribution for $X$.

Calculate the exact value of $p$.

Frank plays the game once.

Calculate the expected score.

Frank plays the game twice and adds the scores together.

Find the probability Frank has a total score of $4$, giving your answer as a fraction.

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

Find the value of $k$.

Find $E\left(X\right)$.

A game is played where contestants shoot a football at a goal with a goal keeper. The goal is divided into five regions: A, B, C, D and E. Each region is assigned a score, $X$, outlined in the table below.

The following table shows the value of $X$ for each region and the probability distribution for $X$.

It is given that $p=2q$.

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

Calculate the expected score.

Find the probability that a player has a score of 16 after two rounds.

A biased coin has a probability of showing tails of 0.85. Leon plays a game where he flips the coin. He pays $15 to play. If the coin lands on tails he receives nothing, but if it lands on heads he receives $5c$ dollars. The game is fair.

Determine the value of $c$ and write down the total prize if he wins.

A weekly raffle ticket costs $k$ dollars, with three different levels of prize, $S$ dollars. The grand prize in the first week is $100 and it increases by $5 every week if nobody wins it.

The following table shows the probability distribution for $S$.

Find the value of $p$.

Given the grand prize is not won, write down an expression for the grand prize, $G$, in the form $G=a+bn$, where $a$ and $b$ are constants to be found and $n$ is the week of the raffle.

Given the raffle is a fair game in the fourth week, find the value of $k$.

The ticket price stays at the value of $k$ found in part (c).

Find the week in which the expected profit for the ticket buyer is $5.

A shooting target is divided into three regions A, B and C. Contestants pay $7.50 to enter and get to take one shot.

The probability of hitting each region is given in the following table.

It is given that $3a=b$ and $a,b\in \mathrm{\mathbb{Z}}$.

Find the value of $a$ and the value of $b$.

A contestant’s prize depends on the region they hit.

Calculate the value of $k$ such that the game is a fair game.

A discrete random variable $X$ 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 $Y$.

Two fair six-sided dice are rolled. One is a standard die with sides numbered 1 to 6 and the other die has sides numbered 1, 1, 2, 2, 4, 4.

The discrete random variable $S$ is the sum of these two dice when they are rolled.

Complete the following probability distribution table.

Find $\mathrm{E}\left(S\right)$.

Let $X$ be the discrete random variable represented in the probability distribution table below.

Find the value of $k$.

Find the expected value of $X$.

The table below represents the number of strokes Josh takes on a particular round of golf and the corresponding frequencies over a year of playing at the same golf course.

Complete the following probability distribution table for the data above.

Par refers to the number of strokes a golfer is expected to need to complete the play on a golf course. The par number of strokes for Josh’s golf course is 72.

Determine whether Josh’s expected number of strokes is less than or greater than the par number of strokes.

The table shows the probability distribution of $X$, the number of devices made by one electronics company that a person living in a city in France owns.

It is given that $p=21q$.

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

Find the expected number of these devices owned by a randomly selected person from this city.

The city has a population of 412 000. On average, each of these devices cost its owner €825.

Estimate the total amount spent on these devices by the people in this city. Give your answer to the nearest euro.

A spinner has six sections; A, B, C, D, E and F.  The table below shows the area of the spinner occupied by each section and their respective pay offs.

Calculate the value of p.

The game costs $4 and John says that the expected profit from playing the game is $0.30.

Calculate the percentage error in John’s claim.

Two biased coins are tossed and a fair spinner divided into four equal sectors numbered 1 to 4 is spun.

Write down the total number of possible outcomes when the two coins are tossed and the spinner is spun.

A random variable, $X$, is defined as the number of tails 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 tails is  $\frac{1}{6}$.

Complete the following probability distribution table for X:

Represent the probability distribution for $X$ as a piecewise function in the form:

$\mathrm{P}(X=x)=f\left(x\right)=\{\begin{array}{ll}& \\ & \end{array}$

Tom has constructed a biased spinner with six sectors labelled 1 to 6.

The probability of the spinner landing on each of the six sectors is shown in the following table.

Find the value of $p$.

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

Tom plays the game twice and adds the two scores together. Find the probability that the sum of Tom’s two scores is 9.

Find the probability that $X$ is

(i) no more than 3,

(ii) at least 5.

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

Find the value of $k$.

Find $E\left(X\right)$.

A weekly lottery ticket costs $15, with five different levels of prizes, $S$ dollars. The grand prize in the first week is $2000 and it increases by 20% each week that nobody wins it.

The following table shows the probability distribution for $S$.

Find the value of $p$.

Determine whether the lottery is a fair game in the first week. Justify your answer.

Given the grand prize is not won, write an expression in terms of $n$ for the value of the grand prize in the $n$th week of the lottery.

The $w$th week is the first week in which a player is expected to make a profit.

Calculate the value of $w$ and the expected profit in that week. Give the expected profit correct to 2 decimal places.

Ben plays a game involving a biased eight-sided die.

The faces of the die are labelled $-4,-2,-1,0,1,3,5,6$.

The score of the game, $X$, is the number which lands face up after the die is rolled.

The following table shows the probability distribution for $X$.

It is given that $p=4q$.

Calculate the exact value of $p$ and the exact value of $q$.

Ben plays the game once.

Calculate the expected score.

Ben plays the game twice and adds the scores together.

Find the probability that Ben has a total score of $-3$, giving your answer as a fraction in its simplest form.

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

It is given that $\mathrm{E}\left(X\right)=2.28$.

Show that $a+b=0.4$ and $a+2b=0.65$.

Hence find the value of $a$ and the value of $b$.

Find $\mathrm{P}(1<X\le 4)$.

The table below represents the number of pets and the corresponding probability of a house having that number of pets.

Find the value of p.

Find the expected number of pets in a house.

There was a recording error and houses with 5 pets were not counted. It is found that there are 6 houses with 5 pets. The neighbourhood in total has 406 houses, including these 6 houses.

Complete the following table for the true probability distribution of the number of pets, x.

Find the actual expected number of pets.

Calculate the percentage error between your answer in part (b) and your answer in part (d).