7.02 Theoretical probability
Amy picks a whole number between 4 and 8 inclusive.
List the sample space.
Find the probability that Amy picked the number 5.
Find the probability that Amy picked the number 1.
Find the probability that Amy picked an even number.
A standard six-sided die is rolled once.
List the sample space.
Find the probability of rolling a 2.
Find the probability of rolling a 6.
Charlene spins the spinner shown:
List the sample space.
Find the probability of spinning a number greater than 29.
Find the probability of spinning a number in the twenties.
Find the probability of spinning a 27 or a 29.
A number is randomly selected from the following list:
\left\{1, 3, 3, 6, 6, 6, 8, 8, 8, 8, 10, 10, 10, 10, 10\right\}
Find the probability of selecting a 1.
Find the probability of selecting a 3.
Find the probability of selecting an 8.
Which number is most likely to be selected?
A 12-sided die has faces with the numbers 1 through 12 as shown:
Find the probability of rolling an odd number.
Find the probability of rolling a multiple of 4.
A year 4 class are sitting at their desks in the order shown below. A student is chosen at random from the class.
| Column 1 | Column 2 | Column 3 | Column 4 | Column 5 | |
|---|---|---|---|---|---|
| Row 1 | Laura | Kenneth | Sophia | Hermione | Valentina |
| Row 2 | Oliver | Maria | Harry | Roald | |
| Row 3 | Gwen | Jenny | Fred | Mario | |
| Row 4 | Aaron | Luke | Elizabeth | Derek |
How many possible outcomes are there?
Find the probability that a student in Column 4 is picked.
Find the probability that a student in the Row 2 is picked.
Find the probability that a student in Year 4 is picked.
A bag contains 17 yellow marbles, 10 grey marbles and 14 orange marbles.
If one marble is picked at random from the bag, find the probability that it is:
Yellow
Grey
Orange
Yellow or orange
Grey or orange
Yellow, grey or orange
A book has pages numbered from 1 to 100. If the book is opened to a random page, find the probability that the page number:
Is a multiple of 9.
Has the digit \rq 6 \rq in the page number.
A cube has six faces, each face is painted a certain colour. If the cube is rolled, find the number of faces that should be painted blue so that the probability of blue appearing on the uppermost face is:
\dfrac{1}{2}
\dfrac{1}{3}
\dfrac{1}{6}
1
Each 26 letters of the alphabet are written on separate pieces of paper and placed in a bag. If one letter is to be picked out of the bag at random find the probability of picking a:
J
K, Y or R
Letter in the word PROBABILITY
M, K, D, O, H or B
Letter in the word WORKBOOK
A bag contains 86 marbles, some of them are black and some are white. If the probability of selecting a black marble is \dfrac{33}{43}, find:
The number of black marbles.
The number of white marbles.
Charlie enters a raffle in which 300 tickets are sold. Find the probability of him winning a prize if he purchases:
1 ticket
2 tickets
3 tickets
n tickets
A circular spinner is divided into three unequal parts. The green sector takes up an angle of 250 degrees at the centre. The red sector takes up an angle of 60 degrees at the centre and the blue sector takes up the remainder of the spinner. Find the probability that the spinner will land on blue.
A six-sided die is rolled. Find:
P \left( \text{odd number} \right)
P \left( \text{number greater than } 1 \right)
P \left( \text{number divisible by } 2 \right)
P \left( \text{number less than } 1 \right)
P \left( \text{neither } 2 \text{ nor } 5 \right)
A marble is randomly drawn from a bag which contains 6 red marbles, 7green marbles and 3 blue marbles. Find:
P \left( \text{red} \right) + P \left( \text{green} \right) + P \left( \text{blue} \right)
P \left( \text{red or green} \right)
P \left( \text{red or blue} \right)
P \left( \text{green or blue} \right)
The sample space of an event is listed as S = \left\{\text{short}, \text{average}, \text{tall}\right\}. If P \left( \text{average} \right)=0.5 and P \left( \text{short} \right)=0.3, find P \left( \text{tall} \right).
For each of the following, state whether the two events are complementary:
Event 1: Selecting a positive number.
Event 2: Selecting a negative number.
Event 1: Drawing a red card from a standard deck of cards (no jokers).
Event 2: Drawing a black card from a standard deck of cards (no jokers).
Event 1: Drawing a club from a standard deck of cards (no jokers).
Event 2: Drawing a spade from a standard deck of cards (no jokers).
Event 1: Rolling a number greater than 3 on a die.
Event 2: Rolling a number less than 3 on a die.
Find each of the following probabilities, find the probablity that the complementary event will occur:
\dfrac{4}{5}
0.64
A biased coin is flipped, with heads and tails as possible outcomes. Calculate P \left( \text{heads} \right) if P \left( \text{tails} \right)=0.56.
A bag contains 34 red marbles and 35 blue marbles. If picking a marble at random, find:
P \left( \text{red} \right)
P \left( \text{not red} \right)
A bag contains 50 black marbles, 37 orange marbles, 29 green marbles and 23 pink marbles. If a marble is selected at random, find the following probabilities:
P \left( \text{orange} \right)
P \left( \text{orange or pink} \right)
P \left( \text{not orange} \right)
P \left( \text{neither orange nor pink} \right)
A number between 1 and 100 inclusive is randomly picked.
State the complement of picking a number greater than 61.
Find the probability that the number picked is greater than 61.
A regular die is rolled. Find the probability of:
Not rolling a 4.
Not rolling a 1 or 5.
Not rolling an even number.
Not rolling an 8.
Not rolling a 1,\,2,\,3,\,4,\,5, or 6.
The 26 letters of the alphabet are written on pieces of paper and placed in a bag. If one letter is picked out of the bag at random, find the probability of:
Not selecting a B.
Not selecting a K, R or T
Selecting a letter that is not in the word PROBABILITY
Not selecting a T, L, Q, A, K or Z
Selecting a letter that is not in the word WORKBOOK
A card is drawn at random from a standard deck. Find the probability that the card is:
A diamond
A spade
Not a heart
From a normal deck of cards, find the probability of:
Selecting a five
Selecting a nine
Not selecting a two
Selecting a black card
Not selecting a black card
The number of movie, concert and musical tickets that are offered as a prize in a raffle are in the ratio 8:19:3.
Calculate the probability that the winner will be given a concert ticket.
The winner doesn’t want to see a musical. Calculate the probability that they get a ticket they want.
A card is selected from a standard deck of cards:
Find the probability of:
Selecting a face card.
Selecting a black nine.
Selecting an odd-numbered black card, not counting ace as a numbered card.
Selecting a red nine.
Not selecting a red three.
Not selecting a queen of clubs
Selecting a ten, jack, queen, king or ace
Selecting the king of diamonds
Not selecting a red ten or black jack
Ben has 3 shirts, each in a different colour: crimson (C), pink (P) and white (W), and 4 ties, each in a different colour: blue (B), grey (G), red (R) and yellow (Y).
How many different combinations are possible?
Find the probability that he is wearing:
A pink shirt and yellow tie.
A pink shirt.
A pink or white shirt.
| C | P | W | |
|---|---|---|---|
| B | C,B | P,B | W,B |
| G | C,G | P,G | G,W |
| R | C,R | P,R | W,R |
| Y | C,Y | P,Y | W,Y |
A player rolls two dice and finds the sum of the numbers on the faces.
List the sample space for the sum of two dice.
Find the probability the dice will sum to five.
Find the probability the dice will sum to an odd number.
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
A player rolls two dice and finds the difference, that is, the largest number minus the smaller number.
List the sample space for the difference of two dice.
Find the probability the dice will have a difference of zero.
Find the probability the dice will have a difference of five.
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | 0 | 1 | 2 | 3 | 4 | 5 |
| 2 | 1 | 0 | 1 | 2 | 3 | 4 |
| 3 | 2 | 1 | 0 | 1 | 2 | 3 |
| 4 | 3 | 2 | 1 | 0 | 1 | 2 |
| 5 | 4 | 3 | 2 | 1 | 0 | 1 |
| 6 | 5 | 4 | 3 | 2 | 1 | 0 |
Twenty-two balls coloured either green or black and numbered 1 to 11 are placed in a bag. The table shows all the possible outcomes that can occur:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Green | \text{G}1 | \text{G}2 | \text{G}3 | \text{G}4 | \text{G}5 | \text{G}6 | \text{G}7 | \text{G}8 | \text{G}9 | \text{G}10 | \text{G}11 |
| Black | \text{B}1 | \text{B}2 | \text{B}3 | \text{B}4 | \text{B}5 | \text{B}6 | \text{B}7 | \text{B}8 | \text{B}9 | \text{B}10 | \text{B}11 |
Find the probability that a ball drawn at random from the bag:
Is black.
Has the number 2.
Has the number 2 or 7.
Has a number 3 or higher.
Is green and has the number 5.
Is green or has the number 5.
In a boardgame a player rolls two six-sided dice but can only move the number of spaces as the given by the largest number rolled. The table shows the possible outcomes:
List the sample space for the maximum of the dice.
Determine the probability a player can move 4 or more spaces.
What is the most likely number of spaces a player can move?
Find the probability a player moves an even number of spaces less than 4.
Moving how many spaces has a probability of 25\%?
Determine the probability a player can move 2 or 5 spaces.
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 | 5 | 6 |
| 2 | 2 | 2 | 3 | 4 | 5 | 6 |
| 3 | 3 | 3 | 3 | 4 | 5 | 6 |
| 4 | 4 | 4 | 4 | 4 | 5 | 6 |
| 5 | 5 | 5 | 5 | 5 | 5 | 6 |
| 6 | 6 | 6 | 6 | 6 | 6 | 6 |
Two dice are rolled, and the combination of numbers rolled on the dice is recorded.
Complete the table of outcomes:
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | 1,1 | 1,2 | ||||
| 2 | 2,1 | |||||
| 3 | ||||||
| 4 | ||||||
| 5 | ||||||
| 6 |
Find the following probabilities for the two numbers rolled:
P(1 and 4)
P(1 then 4)
P(difference =4)
P(product =12)
P(difference \leq 2)
P(difference =3)
P(product =20)
P(difference \leq 1)
If the numbers appearing on the uppermost faces are added, state whether the following are true or false:
A sum greater than 7 and a sum less than 7 are equally likely.
A sum greater than 7 is more likely than a sum less than 7.
A sum of 5 or 9 is more likely than a sum of 4 or 10.
An even sum is more likely than an odd sum.
The following two spinners are spun and the result of each spin is recorded:
Complete the table to represent all possible combinations:
State the total number of possible outcomes.
Find the probability that the spinner lands on a consonant and an even number.
Find the probability that the spinner lands on a vowel or a prime number.
| A | B | C | |
|---|---|---|---|
| 1 | 1,A | ||
| 2 | 2,C | ||
| 3 |
The following spinner is spun and a normal six-sided die is rolled:
| W | X | Y | Z | |
|---|---|---|---|---|
| 1 | 1,W | |||
| 2 | 2,Z | |||
| 3 | ||||
| 4 | ||||
| 5 | 5,X | |||
| 6 |
Complete the table above to represent all possible outcomes.
State the total number of possible outcomes.
Find the probability that the spinner lands on X and the dice rolls a prime number.
Find the probability that the spinner lands on W and the dice rolls a factor of 6.
Find the probability that the spinner doesn’t land on Z or the dice doesn't roll a multiple of 3.
A coin is tossed and a die is rolled simultaneously.
Complete the following table to display the possible outcomes:
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| Heads | \text{H,1} | \text{H,3} | \text{H,4} | \text{H,6} | ||
| Tails | \text{T,2} | \text{T,3} | \text{T,5} |
How many possible outcomes are there?
Determine the probability of rolling a 1.
Determine the probability of rolling an even number.
Determine the probability of tossing a head and rolling an even number.
Box A contains 2 red and 2 blue marbles while Box B contains 1 green, 2 yellow and 1 red marbles. A marble is randomly selected from each box.
Complete the given table to represent all possible outcomes:
Determine the probability of:
Drawing marbles the same colour.
Drawing marbles of different colour.
Drawing a red and a yellow marble.
The draw containing at least one red marble.
| G | Y | Y | R | |
|---|---|---|---|---|
| R | R,G | R,Y | R,R | |
| R | R,Y | R,R | ||
| B | B,G | B,Y | B,R | |
| B | B,Y | B,Y |
The following spinner is spun and a normal six-sided die is rolled at the same time. The product of their respective results is recorded.
Construct a table to represent all possible outcomes.
State the total number of possible outcomes.
Find the probability of an odd product.
Find the probability of rolling a 5 on the dice and scoring an even product.
Find the probability of spinning a 3 on the spinner or scoring a product which is a multiple of 4.
The following two spinners are spun and the sum of their results is recorded:
Spinner 1
Spinner 2
Construct a table to represent all possible outcomes.
State the total number of possible outcomes.
Find the probability that the first spinner lands on an even number and the sum is even.
Find the probability that the first spinner lands on a prime number and the sum is odd.
Find the probability that the sum is a multiple of 4.
Two spinners labelled 1 to 4 are spun simultaneously and the results added:
Spinner 1
Spinner 2
Construct a table to represent all possible outcomes.
List the sample space for the sum of the two spinners.
Determine the probability of:
An even result.
A result greater than 6.
A result less than 5 and even.
A result less than 5 or even.
A two-digit number is formed by spinning the following two spinners with Spinner 1 being the first digit and Spinner 2 being the second digit in the number:
Spinner 1
Spinner 2
Construct a table to represent all possible outcomes.
Determine the probability the number formed:
Is 25.
Is even.
Is prime.
Is divisible by 5.
Is divisible by 5 and odd.
Contains at least one 2.