The following spinner is spun and a normal six-sided die is rolled. The product of their respective results is recorded.
| 2 | 3 | 5 | 8 | |
|---|---|---|---|---|
| 1 | ||||
| 2 | ||||
| 3 | ||||
| 4 | 12 | |||
| 5 | ||||
| 6 | 48 |
Complete the table above to represent all possible outcomes.
State the total number of possible outcomes.
Find the probability of:
An odd product.
A 5 on the dice and scoring an even product.
A 3 on the spinner or scoring a product which is a multiple of 4.
Two dice are rolled, and the combination of numbers rolled on the dice is recorded.
Complete the table of outcomes:
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)
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | 1,1 | 1,2 | ||||
| 2 | 2,1 | |||||
| 3 | ||||||
| 4 | ||||||
| 5 | ||||||
| 6 |
The numbers appearing on the uppermost faces are added. State whether the following are true.
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 sum of their results are added:
Complete the given 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.
The first spinner lands on a prime number and the sum is odd.
The sum is a multiple of 4.
| 7 | 9 | 12 | |
|---|---|---|---|
| 2 | 9 | ||
| 3 | 13 | ||
| 4 |
An archer makes two attempts to hit a target, and the probability that he hits the target on any one attempt is \dfrac{1}{3}. Find the probability that the archer will miss the target on both attempts.
Christa enters a competition in which she guesses the 3-digit code (from 000 to 999) which cracks open a vault containing one million dollars. If the 3-digit number to open the vault is randomly generated by a computer, find the probability that it is:
An odd number.
An even number (including 000).
A number greater than 123.
A number divisible by 10.
A number less than 321.
A fair die is rolled twice. Find the probability of:
Rolling a 6 and a 1 in any order.
Rolling a 6 and then a 1.
A computer generates two random numbers between 1 and 100 (inclusive). Using the product rule, find the probability that the two numbers are:
The same.
Different.
89 and 83 in that order.
89 and 83 in any order.
14 and 36 in any order.
Not 16 and 95.
A computer generates a two-digit number by randomly selecting the first digit from 1, 3, 5, 7 and 9, and randomly selecting the second digit from 0, 2, 4, 6 and 8. Using the product rule, find the probability that number generated is:
94
Even
Odd
Less than 32
Greater than 74
In a game of Monopoly, rolling a double means rolling the same number on both dice. When you roll a double this allows you to have another turn. Find the probability that Sarah rolls:
A double 1.
A double 5.
Any double.
Two doubles in a row.
Two dice are rolled, and the numbers appearing on the uppermost faces are added. Find:
P(a total less than 6)
P(a total greater than or equal to 10)
P(a total of at least 7)
James is in the middle of a game of Monopoly, which involves players rolling two dice at a time. James is desperate to reach the Go! square and collect \$200. Calculate the probability that he lands exactly on the Go! square if it is:
2 squares ahead of his current position.
3 squares ahead of his current position
4 squares ahead of his current position
5 squares ahead of his current position
A pile of playing cards has 4 diamonds and 3 hearts. A second pile has 2 diamonds and 5 hearts. One card is selected at random from the first pile, then the second.
Construct a probability tree of this situation with the correct probability on each branch.
Find the probability of selecting two hearts.
Construct a probability tree showing all the ways a captain and a vice-captain can be selected from Matt, Rebecca, Helen and Chris.
A bag contains the letters A, B, C, D and E. Two letters are randomly drawn out of the bag without replacement.
Construct a probabilty tree showing all possible outcomes.
Find the probability of the following:
A is drawn followed by B.
Two A's are drawn.
C is not drawn.
D is drawn.
E is drawn and B is not.
C or D is drawn.
Neither C nor D are drawn.
Mae deals two cards from a normal deck of cards. Calculate the probability that she deals:
Two 10s
Two red cards
Two diamonds
A 10 of spades and an Ace of diamonds, in that order
A 10 of clubs and an Ace of spades, in any order
Buzz is dealing cards from a regular deck. Find the number of possible outcomes there are if the number of cards being dealt is:
Two
Three
Five
Lucy has a box of Favourites chocolates. In this box there are 30 chocolates, 5 of which are Picnics. Lucy takes and eats a chocolate without looking until she gets a Picnic.
Find the probability she only eats one chocolate.
Find the probability she eats only two chocolates.
Find the probability she eats five chocolates.
As she eats more and more chocolates, is the probability of the next chocolate being a Picnic getting higher, lower or the same?
How many chocolates must Lucy have eaten to be certain that the next chocolate will be a Picnic?
Tara takes a bus to the station and then immediately gets on a train so she can get to work exactly on time. The probability that her bus is on time is 0.5 and the probability that her train is on time is 0.7.
Draw a probability probability tree that shows all possible outcomes and probabilities.
Find the probability that both vehicles are late.
Find the probability that her bus is on time but Tara is still late to work.
If the bus is late, the probability that Tara gets to her train (which is on time) is 0.1. Find the probability that the bus is late and she doesn’t get to her train, given that her train is on time.
A carton contains a dozen eggs, of which 4 contain no yolk. If 3 eggs are chosen at random for a cake, find the probability that they all have no yolk.
Quiana is tossing a coin. She keeps tossing the coin until a Head appears. Her first set of tosses went Tails, Tails, Heads. So she stopped after three tosses. She repeated the experiment 19 more times and recorded her results on the following table:
Based off Quiana's experiment, find the experimental probability that it takes 4 tosses of the coin before a Head appears.
Find the theoretical probability that it takes 4 tosses of a coin before a Head appears.
Was the experimental probability Quiana found, greater or less than the theoretical probability?
| \text{Number of Tosses} \\ \text{before Head appears} | \text{Frequency} |
|---|---|
| 1 | 6 |
| 2 | 4 |
| 3 | 3 |
| 4 | 3 |
| 5 | 4 |
Two standard die are rolled. One is red and one is white. Calculate the probability that:
The same number is rolled.
The sum of the two outcomes exceeds 9.
The red die is 4 and the white die 6.
The red die is even and the white die odd.
The sum of the two outcomes is less than 2.
A class contains 5 girls and 6 boys. Two are selected for a class committee. Find the probability that a girl and boy are selected.
Christa gets to and from school by car, bus or bike.
If she goes to school by bike, she won’t use the bus coming home.
Is she goes to school by bus or car, she won’t cycle home.
Draw a tree diagram illustrating all possible combinations of her to and from journey to school.
State the number of possible outcomes.
If each trip is equally likely, find the probability that:
Christa uses two different forms of transport to and from school.
Christa travels by car and bus.
Christa travels by car or bus.
From a set of 10 cards numbered 1 to 10, two cards are drawn at random without replacement. Find the probability that:
Both numbers are even.
One is even and one is odd.
The sum of the two numbers is 12.
A number game uses a basket with 9 balls, all labelled with numbers from 1 to 9. 3 balls are drawn at random, without replacement. Find the probability that:
The ball labelled 3 is picked.
The ball labelled 3 is picked and the ball labelled 6 is also picked.
Three marbles are randomly drawn without replacement from a bag containing 6 red, 5 yellow, 3 white, 5 black and 6 green marbles. Find the probability of drawing:
3 white marbles.
3 black marbles.
Zero green marbles.
Zero yellow marbles.
At least one red marble.
There are four cards marked with the numbers 2, 5, 8, and 9. They are put in a box. Two cards are selected at random one after the other without replacement to form a two-digit number.
Draw a tree diagram to illustrate all the possible outcomes.
How many different two-digit numbers can be formed?
Find the probability of obtaining:
A number less than 59.
An odd number.
An even number.
A number greater than 90.
A number divisible by 5.
Consider the following four numbered cards:
Two of the cards are randomly chosen and the sum of their numbers is listed in the following sample space:
\left\{15, 10, 8, 11, 9, 4\right\}
Find the missing number on the fourth card.
If two cards are chosen at random, find the probability that the sum of their numbers is:
Even
At least 10
Beth is packing for a holiday when she realises that she only has enough room in her suitcase for two pairs of shoes. She randomly selects two pairs to pack from the 30 pairs of shoes that she owns in total.
If Beth has 10 pairs of high heels, find the probability that she packs at least one pair of these.
If Beth has 6 pairs of running shoes, find the probability that she packs two pairs of these.
Bag A contains 15 red marbles and 10 blue marbles, while bag B contains 20 red marbles and 25 blue marbles. If Eileen is to select a marble by first selecting a bag at random and then selecting a random marble from that bag:
Construct a probability tre showing all of Eileen’s possible selections.
Find the probability that the marble selected is:
From bag A
From bag B
Red
In a game of Blackjack, a player is dealt a hand of two cards from the same standard deck. Find the probability that the hand dealt:
Is a Blackjack. (An Ace paired with 10, Jack, Queen or King.)
Has a value of 20. (Jack, Queen and King are all worth 10. An Ace is worth 1 or 11.)
Nine pilots from StarJet and 7 pilots from AirTiger offer to take part in a rescue operation. If 2 pilots are selected at random:
Construct a probability tre showing all possible combinations of airlines from which the pilots are selected.
Find the probability that the two pilots selected are from:
The same airline
Different airlines
There are 18 M & Ms left in a packet. All are different colours, of which one is blue. Paul picks them one by one, notes the colour and then eats them. But if he picks the blue one, he puts it back into the packet and chooses again.
Find the probability that:
The first one he picks is not blue.
The first two he picks are not blue.
The first one he eats is actually the second one he has picked.
The first one he eats is actually the third one he has picked.
In tennis if the first serve is a fault (out or in the net), the player takes a second serve. A player serves with the following probabilities:
First serve in: 0.55
- Second serve in: 0.81
Construct a probability tree showing the probability of the first two serves either being in or a fault.
Find the probability that the player needs to make a second serve.
Find the probability that the player makes a double fault (both serves are a fault).
A container holds four counters coloured red, blue, green and yellow. Draw a tree diagram representing all possible outcomes when two draws are done, and the first counter is:
Replaced before the second draw.
Not replaced before the next draw.
A container holds three cards coloured red, blue and green.
Draw a tree diagram representing all possible outcomes when two draws occur, and the card is not replaced before the next draw.
Find the probability of drawing the blue card first.
Find the probability of drawing a blue card in either the first or second draw.
Find the probability of drawing at least one blue card if the cards are replaced before the next draw.
Two marbles are randomly drawn without replacement from a bag containing 1 blue, 2 red and 3 yellow marbles.
Construct a tree diagram to show the sample space.
Find the probability of drawing the following:
A blue marble and a yellow marble, in that order.
A red marble and a blue marble, in that order.
2 red marbles.
No yellow marbles.
2 blue marbles.
A yellow marble and a red marble, in that order.
A yellow and a red marble, in any order.
Consider the word WOLLONGONG. If three letters are randomly selected from it without replacement, find the probability that:
The letters are W, O, L, in that order.
The letters are O, N, G, in that order.
All three letters are O.
None of the three letters is an O.
There are 4 green counters and 8 purple counters in a bag. Find the probability of choosing a green counter, not replacing it, then choosing a purple counter.
A standard deck of cards is used and 3 cards are drawn out. Find the probability, in fraction form, of the following:
All 3 cards are diamonds if the cards are drawn with replacement.
All 3 cards are diamonds if the cards are drawn without replacement.
A number game uses a basket with 6 balls, all labelled with numbers from 1 to 6. 2 balls are drawn at random. Find the probability that the ball labelled 3 is picked once if the balls are drawn:
With replacement.
Without replacement.
A hand contains a 10, a jack, a queen, a king and an ace. Two cards are drawn from the hand at random, in succession and without replacement. Find the probability that:
The ace is drawn.
The king is not drawn.
The queen is the second card drawn.
Eileen randomly selects two cards, with replacement, from a normal deck of cards. Find the probability that:
The first card is a queen of spades and the second card is a 4 of clubs.
The first card is spades and the second card is a 4.
The first card is a Queen and the second card is black.
The first card is not a 7 and the second card is not Clubs.
Find the probability of drawing a green counter from a bag of 5 green counters and 6 black counters, replacing it and drawing another green counter.
A chess player is placed into a draw where in each match he has a 30\% chance of winning. Find the probability that:
He wins his first two matches.
He wins his first three matches.
He wins his first two matches and then loses his third match.
Beth randomly selects three cards, with replacement, from a normal deck of cards. Find the probability that:
The cards are queen of diamonds, king of spades, and king of diamonds, in that order.
The cards are all black.
The first card is a 9, the second card is a heart and the third card is red.
The cards are all hearts.
None of the cards is a 9.
From a standard pack of cards, one card is randomly drawn and then put back into the pack. A second card is then drawn. Calculate the probability that:
Neither of the cards are diamonds
At least one of the cards is a diamond
Three marbles are randomly drawn with replacement from a bag containing 6 red, 4 yellow, 3 white, 2 black and 4 green marbles. Find the probability of drawing:
Three white marbles
No green marbles
At least one red marble
At least one white marble
Amelia randomly selects two cards, with replacement, from a normal deck of cards. Find the probability that:
Both cards are red
Both cards are the same colour
Both cards are of different colours
James, a test cricketer, analysed his past innings and found his probabilities of scoring particular numbers of runs, as shown in the table:
| \text{Number of runs} | 0 | 1 - 20 | 21 - 49 | 50 - 99 | 100+ |
|---|---|---|---|---|---|
| \text{Probability} | \dfrac{1}{10} | \enspace \enspace \dfrac{3}{10} | \enspace \enspace \dfrac{3}{10} | \enspace \enspace \dfrac{2}{10} | \enspace \dfrac{1}{10} |
If James is selected to play in the next test match, calculate the probability that he scores:
0 runs in exactly one of the two innings.
0 runs in both innings.
At least 50 runs in at least one of the innings.
A total of 100 or more runs in each of the two innings.
More than 20 runs in only one of the two innings.
Based on historical data, it rains 19 days out of 31 in July in Bangkok, Thailand.
Find the chance it will rain in Bangkok on the 7th July 2027.
Find the probability, to the nearest percent, that:
It will rain two days in a row.
From a Monday to a Wednesday in July, it only rained on the Monday.
From a Monday to a Wednesday in July, it rained on only one of the days.
In a school, 25\% of students ride skateboards and 20\% of students have dark hair. One student is selected at random. Find the probability that the student:
Has dark hair and rides a skateboard.
Has light hair and does not ride a skateboard.
Has dark hair and does not ride a skateboard.
Has light hair and rides a skateboard.
The ratio of left-handed people to right-handed people in a country is 4:3. Two people are surveyed at random. Calculate the probability that:
Both people are left-handed.
One person is left-handed and the other is right-handed.
At least one person is right-handed.