Yvonne is struggling to decide what to watch online. She decides to pick one movie at random from the streaming website. A Venn diagram of her options sorts movies into three categories based on their genre: Comedy, Action and Horror:
How many of the movies are horror films?
How many of the movies fit into only one genre?
Find the probability of selecting a film that fits into only one genre given that it fits into the horror genre.
Find P\left( \left. A \right| B \right) given the following probability Venn Diagram:
Given that and A and B are independent events such that P\left(A\cap B\right) =0.2 and P\left(A\cap B' \right) =0.3, find the value of:
P \left( A \right)
P \left( B \right)
P \left(A\cup B' \right)
Sally has 10 songs in a playlist. Five of the songs are her favourite. She starts to play them on shuffle so each song will be played once until all songs have been played. Find the probability that:
The first song is one of her favourites
Two of her favourite songs are the first to be played.
At least 1 of her favourite songs is played in the first three.
The probability that Vanessa, Nadia and Adam get permission to go on their school trip are 0.8, 0.1 and 0.2, respectively. Find the probability that at least 1 of them gets permission.
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.
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 cards marked with the numbers 2, 5, 8, and 9 that were put in a box. Two cards are selected at random one after the other without replacement to form a 2-digit number.
How many different 2-digit numbers can be formed?
Find the probability of obtaining a number that is:
Odd
Even, given that one of the numbers selected is 2
Greater than 90, given that the second number drawn was a 5.
A school consisting of 410 primary students and 410 secondary students offers everyone an optional end of year project to improve their grade. There are various types of projects they can do.
A summary of the number of student submissions are shown in the given table:
| Essay | Creative story | Art work | Other Project | Did not submit | |
|---|---|---|---|---|---|
| Primary | 18 | 64 | 46 | 24 | 258 |
| Secondary | 76 | 68 | 73 | 45 | 148 |
Find the probability of a selecting a student that submitted a project.
Find the probability of a selecting a student that submitted a project given that they are in a secondary year (S).
Find the probability that a submitted project is an essay.
Find the probability that a submitted project is an essay given that the project was submitted by a primary student (P).
In a survey, 600 people were questioned as to whether they read novels in paperback form or on an e-reader:
20\% said they do both.
30\% said they read paperbacks.
The probability that a person who didn’t read paperbacks also didn’t read on an e-reader was 25\%.
Complete the two-way table for the number of people in each category:
| Paperback | Not Paperback | Total | |
|---|---|---|---|
| E-Reader | |||
| Not E-Reader | 60 | ||
| Total |
Find the probability that a randomly selected person reads novels on an e-reader but not in paperback.
Of those that read on e-readers,calculate the percentage read in both formats. Round your answer correct to the nearest percent.
A sample of 20 students, A - T, from the same school were asked whether they had a haircut in the last month (H) and whether they had been sick in the last month (S).
The results are shown in the given tables:
| Student | A | B | C | D | E | F | G | H | I | J |
|---|---|---|---|---|---|---|---|---|---|---|
| \text{Hair} | ✓ | ✓ | - | ✓ | - | ✓ | ✓ | - | - | ✓ |
| \text{Sick} | - | ✓ | - | ✓ | - | - | ✓ | ✓ | - | - |
| Student | K | L | M | N | O | P | Q | R | S | T |
|---|---|---|---|---|---|---|---|---|---|---|
| \text{Hair} | - | - | - | ✓ | - | ✓ | - | ✓ | - | - |
| \text{Sick} | ✓ | ✓ | ✓ | ✓ | - | - | ✓ | ✓ | - | ✓ |
Find the probability that a student:
Had a haircut in the last month.
Had been sick in the last month.
Was sick and had a haircut in the last month.
Find the value of P \left( H \right) \times P \left( S \right).
Does the data suggest that getting haircuts and being sick in the last month are independent or dependent?
Students were asked what they are allergic to. The table shows the results:
| Allergic to nuts | Not allergic to nuts | |
|---|---|---|
| Allergic to dairy | 14 | 12 |
| Not allergic to dairy | 25 | 25 |
If a student is chosen at random, find the probability that:
The student is allergic to dairy.
The student has an allergy.
The student does not have an allergy?
The student is also allergic to dairy if he or she is allergic to nuts.
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.
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).
In a survey, 100 office workers are asked about whether they buy a coffee (set C) or lunch (set L) during the work day. Some of the results are shown in the given Venn diagram. Find:
n \left( L \cap C\rq \right)
P \left( \left. C \right| L \right)
P \left( \left. L \right| C\rq \right)
Of the 100 students starting kindergarten surveyed, 80\% believe in Santa, 41\% believe in the Tooth Fairy and 31\% believe in both.
Complete the following table:
| Believe in Santa | Do not believe in Santa | Total | |
|---|---|---|---|
| Believe in Tooth Fairy | |||
| Do not believe in Tooth Fairy | |||
| Total | 100 |
Of those who believe in Santa, what percentage also believe in the tooth fairy? Round your answer to two decimal places.
Of those who believe in either Santa or the Tooth Fairy, what fraction believe in both?
What proportion of those who believe in the Tooth Fairy do not believe in Santa?
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
In a memory game, 16 pairs of identical cards are randomly placed face down. When someone has a go, they turn one card over and then turn a second card over to try to find an identical pair. If an identical pair are found, they are removed from the pack.
At the beginning of the game, Kathleen turns a card over. Find the probability that she will pick a card that will make an identical pair.
The first two cards Kathleen picks are identical. Is the probability of picking any more pairs of identical cards independent or dependent on the number of pairs picked before?
Two sets of numbers, A and B, are such that set A contains the even numbers from 1 to 20, inclusive, and set B contains the factors of 48 from 1 to 20, inclusive.
List the numbers in set A.
List the numbers in set B.
Find the total number of unique numbers across both sets.
Find the probability that a randomly selected number is in set B, given that it is in set A.
In a survey, 87 people are questioned about whether they own a tablet \left( T \right) or a smartphone\left( S \right). The following probabilities were determined from the results:
P \left( \left. T \right| S \right) = \dfrac{5}{12}, \quad P \left( S \cap T\rq \right) = \dfrac{35}{87}, \quad P \left( T \right) = \dfrac{14}{29}Find n \left( S \cap T \right).
Find P \left( S\rq \cap T \right).
Find P \left( \left. S \right| T \right).
Find P \left( \left. T \right| S\rq \right).
Outcomes
VCMSP347
Describe the results of two- and three-step chance experiments, both with and without replacements, assign probabilities to outcomes and determine probabilities of events. Investigate the concept of independence.
VCMSP348
Use the language of ‘if ....then, ‘given’, ‘of’, ‘knowing that’ to investigate conditional statements and identify common mistakes in interpreting such language.