Probability!(PB1)! RelativeFrequency!and!!...

1 General Mathematics (Preliminary Course) | Relative Frequency and Probability (PB1)

Probability (PB1)

Relative Frequency and Probability

Name .......................................................................

G. Georgiou


An outcome is ……………………………………………………………………………………………………………. …………………………………………………………………………………………………………………………………… Sample space is ………………………………………………………………………………………………………. …………………………………………………………………………………………………………………………………… An event is ………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… Identify the sample space of the following events. (a) Rolling a die. …………………………………………………………………………………………………………………………………… (b) Choosing a marble from a bag of 3 yellow and 4 red marbles. …………………………………………………………………………………………………………………………………. (c) Choosing a letter from the word “PYTHAGORAS”. …………………………………………………………………………………………………………………………………… How many possible outcomes are there: (a) If you choose a letter from the alphabet? …………………………………………………………………………………………………………………………………… (b) If you choose a single digit number? …………………………………………………………………………………………………………………………………… (c) If you pick a marble from a bag which only contains red marbles? ……………………………………………………………………………………………………………………………………

Example 1

Example 2


A student has a bag that contains the letters of the words “HYPOTENUSE”. (a) List the sample space. …………………………………………………………………………………………………………………………………… (b) How many outcomes are there? …………………………………………………………………………………………………………………………………… (c) List the outcomes in each of the following events for the above scenario.

(i) Choosing a vowel. …………………………………………………………………………………………………………………………………… (ii) Choosing the letter “G”. …………………………………………………………………………………………………………………………………… (iii) Choosing a letter after S in the alphabet. …………………………………………………………………………………………………………………………………… These two spinners are both divided into five segments. Each segment is numbered.

If the two spinners are spun and the numbers are added, how many items in the sample space?

…………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………

Example 3

Example 4


A jar contains seven blue marbles and five red marbles. Tony takes a marble from the jar and does not replace it. Tim then takes a marble from the jar. If it is the same colour as Tony's marble, Tim wins the jar of marbles. Which statement is correct? A) Tim is more likely to win if Tony has taken a red marble.

B) Tim is more likely to win if Tony has taken a blue marble. C) The colour that Tony takes has no effect on the likelihood of Tony winning. D) Tim would be more likely to win if they started out with an equal number of each colour.

…………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………

Example 5


• Identify Events with Equally Likely Outcomes When we toss a standard coin, the chance of the coin landing on heads or tails is even. In other words, both outcomes of this event are equally likely. The same can be said when we roll a standard die; the chance of getting a 1 is the same as the chance of getting a 2 or any other number less than or equal to 6. List two other events where the outcomes are equally likely to occur. 1. ……………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………… 2. ……………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………. Are all the outcomes in the following events equally likely? Justify your response. (a) Picking a marble from a bag that contains 3 red and 4 yellow marbles. …………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………….. ………………………………………………………………………………………………………………………………….. (b) Picking a picture card from a standard deck of cards. …………………………………………………………………………………………………………………………………. …………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………….. (c) Choosing a letter from the word “GEORGIOU”. …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………..

Example 6

Example 7


• Using the Following Definition of the Probability of an Event where Outcomes are Equally Likely:

P (event) = !"#$%&'()')*+("&*$,%'("-.(#%/-(-*,'!"#$%&'()'("-.(#%/

If you roll a standard die, calculate: (a) P (2) …………………………………………………………………………………………………………… (b) P (even) …………………………………………………………………………………………………………… (c) P (<4) …………………………………………………………………………………………………………… (d) P (<7) …………………………………………………………………………………………………………… The likelihood of an event can be described using words, such as impossible, unlikely, even chance, likely and certain. These words can be matched up with a corresponding number between 0 and 1. The probability of an …………………………………………………….. event has a probability of 0. The probability of a …………………………………………………….. event has a probability of 1. The probability of all other events can be found more accurately using the formula above.

Formula

We can find the probability of an event where all the outcomes are equally likely by using the following formula:

P (event) = !"#$%&'()')*+("&*$,%'("-.(#%/-(-*,'!"#$%&'()'("-.(#%/

PROVIDED ON HSC FORMULA SHEET

Example 8


Cards with the numbers 101-‐120 are cut up and placed in a bag. (a) Calculate the probability of selecting a card greater than 105. …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… (b) Calculate the probability of selecting a card with a 0 as one of the digits. …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… (c) Calculate the probability of selecting a card greater than 120. …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… (d) Calculate the probability of selecting an even number. …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… (e) Place the events (a) – (d) on the number line below.

Impossible (0) Even chance (0.5) Certain (1)

Example 9


• Calculating Probabilities in terms of Fractional, Decimal or Percentage Chance • Demonstrating the Range of Possible Probabilities, 0 ≤ P (event) ≤ 1, through

Examination of a Variety of Results The probability of an event is generally expressed as a simplified fraction. However, it can also be expressed as a percentage or a decimal. A bag contains 10 marbles. 3 are red, 4 are yellow, 2 are black and 1 is white. Calculate the following probabilities, expressing your answer as a fraction, decimal and a percentage. (a) P (red) …………………………………………………………………………………………………………………………………… (b) P (white) …………………………………………………………………………………………………………………………………… (c) P (green) …………………………………………………………………………………………………………………………………… (d) P (yellow or black) …………………………………………………………………………………………………………………………………… (e) P (yellow, or black or white, or red) …………………………………………………………………………………………………………………………………… (f) Place the probability of the events above on the number line below.

Impossible (0) Even chance ( !")

Certain (1)

Example 10


(g) Is it possible for the probability of an event to be less than zero? Explain your answer. …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… (h) Is it possible for the probability of an event to be greater than 1? Explain your answer. …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… NOTE: Every probability must lie between 0 and 1 (0 ≤ P (event) ≤ 1) A jar contains chocolate chip cookies and Anzac biscuits. If Jim makes a selection at

random from the jar, he has a !" probability of selecting an Anzac biscuit. If there

are 10 chocolate chip cookies, how many Anzac biscuits are there? …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………

Example 11


Whilst not commonly used in H.S.C. examinations, it is important for you to understand the components of a deck of cards.

In a deck of cards, there are:

• 52 cards: 26 of which are red and 26 of which are black. • 4 suits (hearts, diamonds, clubs, and spades). • 13 cards are in each suit. • Picture cards called Jack (J), Queen (Q) and King (K). • Number cards called Ace (A), 2, 3, 4, 5, 6, 7, 8 and 9.

Calculate the probability of: (a) Selecting a picture card from a standard deck of cards. …………………………………………………………………………………………………………………………………… (b) Selecting a red card from a standard deck of cards. …………………………………………………………………………………………………………………………………… (c) Selecting a heart from a standard deck of cards. …………………………………………………………………………………………………………………………………… (d) Selecting a jack of hearts from a standard deck of cards. …………………………………………………………………………………………………………………………………… (e) Selecting a red ace from a standard deck of cards. …………………………………………………………………………………………………………………………………… Activity Ex 5.01 ALL

Example 12


• Determining the Number of Outcomes for a Multi-‐Stage Event by Multiplying the Number of Choices at each Stage

• Using Systematic Lists to Verify the Total Number of Outcomes for Simple Multi-‐Stage Events

Ever wondered why the RTA changed the standard number plate from having 3 letters and 3 numbers to having 2 letters, 2 numbers and then 2 letters? Or why in 1996, an extra digit was added to our phone numbers? The answer lies in examining the amount of combinations possible. To determine the number of possible outcomes for a multi-‐stage event, simply multiply the number of possible outcomes at each stage of the event. A standard coin is tossed twice. (a) Calculate the number of possible outcomes …………………………………………………………………………………………………………………………………… (b) Use the table below to assist you in listing all the possible outcomes.

Fred and Dianne go out for dinner one night. The menu has 4 starters, 12 mains and 5 desserts. If Fred and Dianne were to eat a meal from every course, how many possible combinations of dinner can they eat? …………………………………………………………………………………………………………………………………... …………………………………………………………………………………………………………………………………... …………………………………………………………………………………………………………………………………...

Example 13

Example 14


Jade tosses a coin and then throws a die. (a) How many outcomes are possible for each stage? ………………………………………………………………………………………………………………………………… (b) Use a table to assist you to list the number of outcomes for this experiment. NSW postcodes begin with a 2 and are followed by three digits. (a) Calculate the number of possible postcodes. …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… (b) What could Australia Post do if they needed more postcodes? …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… (c) Australia Post has had a problem with its scanners and their ability to read digits. The solution to this problem is to make every digit different from the other. How many postcodes are possible if the postcode must begin with 2 but every postcode must be unique? …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………

Example 15

Example 16


Explain why either local or federal governments were forced to: (a) Change standard NSW number plates from having 3 letters and 3 numbers to having 2 letters, 2 numbers, and then 2 letters. Use mathematical calculations to justify your answer. …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… (b) Change phone numbers from having 7 digits to having 8 digits (not including area codes). Use mathematical calculations to justify your answer. …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………

Activity Ex 5.02 Q 1, 2, 3, 6, 7, 9 Ex 5.03 Q 1, 3, 4, 6, 7, 9-‐13, 15, 17

Example 17


Morse code is a way of transmitting telegraphic messages. Morse code has developed a way of writing every letter and number in the English alphabet. Each letter is composed of a combination of dots or dashes. This combination never exceeds more than 5 characters. Some examples include:

A . – B -‐ … 1 . -‐ -‐ -‐ -‐ 2 . . -‐ -‐ -‐ 1. How many different possible combinations are there in Morse Code if: (a) Only one character is used? …………………………………………………………………………………………………………………………………… (b) Only two characters are used? …………………………………………………………………………………………………………………………………… (c) Only three characters are used? …………………………………………………………………………………………………………………………………… (d) Only four characters are used? …………………………………………………………………………………………………………………………………… (e) Only five characters are used? …………………………………………………………………………………………………………………………………… 2. All together, how many letters, numbers or symbols can be represented using Morse Code? …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………

Example 18


• Comment Critically on the Validity of Simple Probability Statements Briefly explain why each of the following statements is incorrect.

(a) The probability of driving to a set of traffic lights and the light being red is !"

because the lights can be three colours and red is one of them. …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………

(b) The probability of a name beginning with X is !"#

.

…………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………

(c) The probability of a car accident happening as you drive is !".

…………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………

(d) The probability of living in Australia is !"!#

because there are 218 countries in

the world. …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………

Example 19


• Calculate the Probability of the Complement of an Event using the Relationship P (an event does NOT occur) = 1 – P (an event does occur)

Consider the following scenario: A bag contains 3 red marbles and 7 yellow marbles. We say the complement to selecting a red marble is selecting a yellow marble. In other words, it is the other possibility. Write down the complement of the following events. (a) Rolling an even number on a standard die. …………………………………………………………………………………………………………………………………… (b) Picking a yellow sock from a draw containing yellow and white socks. …………………………………………………………………………………………………………………………………… Answer the following questions: (a) Calculate the probability of rolling a 5 on a standard die. …………………………………………………………………………………………………………………………………… (b) Calculate the probability of not rolling a 5. …………………………………………………………………………………………………………………………………… Notice that your answer to (b) is 1 minus your answer in (a).

Example 20

Example 21

Formula

To find the probability of a complementary event:

P (an event does NOT occur) = 1 – P (an event does occur) !!""=#!!!""

NOT PROVIDED ON HSC FORMULA SHEET


Calculate the probability of the following: (a) Rolling a 4 on a standard die. …………………………………………………………………………………………………………………………………… (b) Not rolling a 4 on a standard die. …………………………………………………………………………………………………………………………………… The digits 0-‐9 are written on cards and placed in a bag. Calculate the following: (a) P (3) ……………………………………………………………………………………………………………………………………

(b) P (! ) …………………………………………………………………………………………………………………………………… (c) P (even) – Note: 0 is not an even number ……………………………………………………………………………………………………………………………………

(d) P (!"!# ) ……………………………………………………………………………………………………………………………………

(e) P (!"#$% ) ……………………………………………………………………………………………………………………………………

Example 22

Example 23


A bag contains 4 red marbles and 5 yellow marbles. (a) Calculate the probability of selecting a red marble. …………………………………………………………………………………………………………………………………… (b) Is selecting a yellow marble the complement to selecting a red marble? …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… (c) Hence calculate the probability of selecting a yellow marble. …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… NOTE: The sum of the probability of an event and the probability of its complement is 1.

Activity Ex 5.04 ALL

Example 24


• Perform Simple Experiments and Use Recorded Results to obtain Relative Frequencies

• Estimate the Relative Frequencies of Events from Recorded Data • Use Relative Frequencies to obtain Approximate Probabilities • Illustrate the Results of Experiments using Statistical Graphs and Displays

(DS2 Displaying and interpreting single data sets)

Relative Frequency is ……………………………………………………………………………….……………….. …………………………………………………………………………………………………………………………………… Jade flipped a coin and achieved the following results.

Heads Tails 22 28

(a) Calculate the relative frequency of the coin landing on heads. …………………………………………………………………………………………………………………………………… (b) Calculate the theoretical probability of a coin landing on heads. …………………………………………………………………………………………………………………………………… (c) Why is the relative frequency different to the theoretical probability? …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… Twister is a board game where a spinner is spun. The spinner has been provided below. We are going to play the game of twister and you are to record how many times the spinner lands in each category.

Example 25

Example 26


(a) Complete the following table:

Tally Relative Frequency

Left Foot (Red)

Left Foot (Blue)

Left Foot (Green)

Left Foot (Yellow)

Left Hand (Red)

Left Hand (Blue)

Left Hand (Green)

Left Hand (Yellow)

Right Foot (Red)

Right Foot (Blue)

Right Foot (Green)

Right Foot (Yellow)

Right Hand (Red)

Right Hand (Blue)

Right Hand (Green)

Right Hand (Yellow)

Total


The table below shows the results of an experiment where fish in a small pond were randomly caught and then released. The colour of each fish was recorded.

From this experiment, Ames found that the experimental probability of randomly selecting a blue fish was 0.3. How many blue fish were caught?

…………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………

Ten coins were tossed. The relative frequency of heads was 1. Which statement is correct? A) All the coins showed heads B) All the coins showed tails C) The coins showed a mixture of heads and tails …………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………

Fish colour Frequency

Red 15

Blue ?

Gold 6

Example 27

Example 28


Records were kept of attempts at goal in a basketball game. Choose the statement that cannot be true. A) Shaq achieved free throws with a relative frequency of 55% B) Lauren achieved free throws with a relative frequency of 1.1

C) Erin achieved free throws with a relative frequency of !"#$

D) Dwayne achieved free throws with a relative frequency of 0 …………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………

Activity Ex 5.05 Q 1-‐6

Example 29

Example 30


• Compare Theoretical Probabilities with Experimental Estimates We can calculate probability using two methods: 1. We can use the formula:

P (event) = !"#$%&'()')*+("&*$,%'("-.(#%/-(-*,'!"#$%&'()'("-.(#%/

2. We can perform simple experiments and record the results to predict future outcomes. When we use method 1, we call this the …………………………………………………………………………………………………………………………………… This is what should happen in theory. When we use method 2, we call this the …………………………………………………………………………………………………………………………………… This allows us to approximate the probability of an event and hence obtain the …………………………………………………………………………………………………………………………………… It is important to note the …………………………………………………………………………………………………………………………………… of an event will vary every time someone carries out that experiment.


(a) Use the twister spinner on page 19 to calculate the theoretical probability of: i. Spinning a right hand red. …………………………………………………………………………………………………………………………………… ii. A right foot. …………………………………………………………………………………………………………………………………… iii. Yellow. …………………………………………………………………………………………………………………………………… (b) Use the relative frequencies from the table on page 20 to calculate the experimental probability of: i. Spinning a right hand red. …………………………………………………………………………………………………………………………………… ii. A right foot. …………………………………………………………………………………………………………………………………… iii. Yellow. …………………………………………………………………………………………………………………………………… (c) Explain why the theoretical probability values are different to the experimental probability values. …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………

Example 31


David rolls two dice and calculates the difference between the largest and smallest number. He records the data in the table below.

Difference Frequency 0 5 1 7 2 8 3 10 4 3 5 2

(a) Calculate how many times David rolled the dice. …………………………………………………………………………………………………………………………………… (b) Calculate the relative frequency of getting a difference of 3. …………………………………………………………………………………………………………………………………… (c) Calculate the experimental probability of getting a difference of 3. …………………………………………………………………………………………………………………………………… (d) Display the information above in a combined frequency histogram and polygon.

Activity Ex 5.07 Q1, 4, 5, 6, 7

Example 32


Question 1 Insert: sometimes, always or never (a) The sum of the probabilities of a pair of complementary events is ________________ 1.

(b) The probability of an event happening is _________ greater than 1.

(c) The probability of an unlikely event is ___________ less than 0.5

(d) Probabilities are _________________ written as fractions and decimals.

(e) The experimental probability and theoretical probability of an event are ______________ equal. Question 2

Explain the difference between the following two symbols P(E) and P(! ). …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… Question 3 Explain the difference between relative frequency and experimental probability. …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… Question 4 Explain what the following mathematical expression means “0!!"#$!% ”. …………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………… Question 5 Explain the difference between a sample space and an outcome. …………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………

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Probability!(PB1)! RelativeFrequency!and!!...

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