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SLIDE 1
Assuming that each of these situations is unbiased, find the probability of
? Rolling a six on a regular six-sided die
? Drawing a Queen of Hearts from a pack of 52 playing cards
? Winning a raffle with one ticket, when 1000 tickets each with a different number have been printed
The word unbiased means that nothing has been done to make one event (outcome) more likely than another. For example, a die has not been tampered with, cards in a pack have not been marked, and so forth. Then each side of the die, or each card in the back, is equally likely to come up, as the choice is random.
SLIDE 2
? Probability of rolling a six on a regular six-sided die is
? Probability of drawing a Queen of Hearts from a pack of 52 playing cards is
? 1000 tickets each with a different number have been printed. Probability of winning with one ticket is
SLIDE 3
? There are 60 boys and girls in total a class of which 23 are boys. If one student is chosen from this class at random, what is the probability of that student being a girl?
? There are 12 counters in a bag. 5 of the counters are blue, 3 are red and the remainder are white. What is the probability of drawing a white counter from the bag?
SLIDE 4
In both cases we assume that every outcome (event) is equally likely.
? There are 60 boys and girls in total a class of which 23 are boys. If one student is chosen from this class at random, what is the probability of that student being a girl?
Solution
There are 37 girls in the class
Probability of a girl
? There are 12 counters in a bag. 5 of the counters are blue, 3 are red and the remainder are white. What is the probability of drawing a white counter from the bag?
Solution
There are 4 white counters in the bag
Probability of a white counter
SLIDE 5
An ordinary fair six-sided die is thrown 120 times.
Estimate the number of times you expect a four to appear.
SLIDE 6
An ordinary fair six-sided die is thrown 120 times. Estimate the number of times you expect a four to appear.
Solution
As we are told that the die is fair, the estimate is based on the probability.
Estimate
SLIDE 7
Probability by experiment
Example
A biased die is thrown 100 times. The number 5 appears 43 times. What is the probability of the number 5 coming up when this die is thrown again?
The die is biased means that something has been done to it to make it more likely that one number will appear than another. When all outcomes are equally likely the situation is called fair. In this example, the situation is biased or unfair.
When the probability is not known in advance for any reason, we estimate probability of an event from a relative frequency found by experiment.
Solution
The relative frequency
We estimate the probability as
We do not know for certain what the probability is, but the result of the experiment is the best choice we have for the answer to this question.
SLIDE 8
? A biased die is thrown 1000 times. A six comes up 45 times. A five comes up 80 times. Estimate the probability of (a) a six, (b) a five, (c) any other number.
? A player in a card game suspects that the cards are marked. During each of forty games, another player has an ace twenty times. Estimate the probability of that player having an ace, and by comparing this with the expected probability should the cards be unbiased, comment whether there is evidence that the cards are marked.
SLIDE 9
? A biased die is thrown 1000 times. A six comes up 45 times. A five comes up 80 times. Estimate the probability of (a) a six, (b) a five, (c) any other number.
(a) Probability of a six
(b) Probability of a five
(c) Probability another number
? A player in a card game suspects that the cards are marked. During each of forty games, another player has an ace twenty times. Estimate the probability of that player having an ace, and by comparing this with the expected probability should the cards be unbiased, comment whether there is evidence that the cards are marked.
Estimate of probability of ace
There are four aces in a pack of 52 playing cards.
Unbiased estimate
The experimental estimate is so much bigger than the unbiased estimate that there is indeed evidence that the cards are marked (or have been tampered with, or the other player is cheating).
SLIDE 10
Total probability
When we throw an ordinary fair six-sided die there are just six possible events in all. Each event has an equal chance of coming up, and the probability of each one is the same, 1/6. Adding up all the probabilities of all the possible events gives a sum of 1. The total probability is 1. The probability of each event may be given in a probability table.
Event 1 2 3 4 5 6
Probability
SLIDE 11
Law of total probability
If all the events are given, then the sum of the probability of all the events is 1.
Question
A biased four-sided die has the following probabilities
Event 1 2 3 4
Probability
What is the probability of throwing a 4? Complete the probability table.
SLIDE 12
The probability of a 4 is
Event 1 2 3 4
Probability
SLIDE 13
The table shows the probabilities that a biased die will land on 1, 2, 3, 4 or 5. If the die is rolled 600 times, what is the best estimate for the number of times it will land (a) on a 6? (b) on a 4, 5 or 6.
Number 1 2 3 4 5 6
Probability 0.1 0.2 0.3 0.15 0.05
SLIDE 14
Number 1 2 3 4 5 6
Probability 0.1 0.2 0.3 0.15 0.05 0.2
(a) Probability land on a 6 .
Estimate
(b) Probability land on 4, 5 or 6
Estimate
SLIDE 15
There are only blue counters, green counters, red counters and black counters in a bag. The table shows the probability of getting a blue counter or a green counter or a red counter. (a) What is the probability of getting a black counter? (b) What is the least possible number of counters in the bag?
Colour blue green red black
Probability 0.25 0.3 0.4
SLIDE 16
Colour blue green red black
Probability 0.25 0.3 0.4 0.05
Least counters 5 6 8 1
The probability of a black counter is
As the probability is there must be at least 20 counters in the bag.
Answers (a) 0.05 (b) 20
SLIDE 17
A book shop had 40 books on a shelf 22 of which were copies of How to Win Friends and Influence People by Dale Carnegie. The rest of the shelf contained copies of Motivation and Personality by Abraham Maslow. After 10 books were sold, the probability of picking a copy of Maslow’s book at random was 50%. How many copies of Motivation and Personality were sold?
SLIDE 18
A book shop had 40 books on a shelf 22 of which were copies of How to Win Friends and Influence People by Dale Carnegie. The rest of the shelf contained copies of Motivation and Personality by Abraham Maslow. After 10 books were sold, the probability of picking a copy of Maslow’s book at random was 50%. How many copies of Motivation and Personality were sold?
Solution
Before the sale, there were 18 copies by Maslow. 10 books were sold, so there were 30 remaining, of which 15 were by Maslow since .
Answer: 3 copies of Maslow’s book were sold, since .
SLIDE 19
Five ladies took part in a singing competition. The table shows the proportion of votes cast for each of the singers. If there were 800,000 votes cast in all, how many voted for Emma?
Singer Alison Beth Charlotte Daisy Emma
Proportion 0.45 0.2 0.15 0.1
SLIDE 20
Five ladies took part in a singing competition. The table shows the proportion of votes cast for each of the singers. If there were 800,000 votes cast in all, how many voted for Emma?
Singer Alison Beth Charlotte Daisy Emma
Proportion 0.45 0.2 0.15 0.1 0.1
Probability voting for Emma = 0.1.
SLIDE 21
A bag contains only red, blue and green counters. The ratio of the number of red counters to blue counters is . A single counter is taken from the bag. The probability that the counter is green is 0.2. What is the probability of taking a blue counter?
SLIDE 22
A bag contains only red, blue and green counters. The ratio of the number of red counters to blue counters is . A single counter is taken from the bag. The probability that the counter is green is 0.2. What is the probability of taking a blue counter?
Solution
Suppose there are 13 red and 3 blue counters in the bag, then there are 16 red and blue counters altogether. This number corresponds to a probability of 0.8.
Then there must be 20 counters in the bag in total, of which 4 are green. So, the probability of taking a blue counter is 3/20.
SLIDE 23
A game uses tiles which are coloured blue, yellow or green only. There are twice as many blue tiles as yellow tiles and four times as many green tiles as blue tiles. If a single tile is taken at random what is the probability that the tile is green?
SLIDE 24
A game uses tiles which are coloured blue, yellow or green only. There are twice as many blue tiles as yellow tiles and four times as many green tiles as blue tiles. If a single tile is taken at random what is the probability that the tile is green?
Solution
The least numbers of tiles are
blue : yellow : green
Total number of tiles
Then the probability of a green counter is
SLIDE 25
In a game there are 100 cards with letters of the alphabet printed on them. The probability of choosing a card with the letter Z is . Then 25 of the cards were lost, after which the probability of choosing a Z was just . How many cards with the letter Z were lost?
SLIDE 26
In a game there are 100 cards with letters of the alphabet printed on them. The probability of choosing a card with the letter Z is . Then 25 of the cards were lost, after which the probability of choosing a Z was just . How many cards with the letter Z were lost?
Solution
Originally, there were cards with the letter Z. When 25 cards were lost, only 75 remained. Since the probability of a Z is then , of these 15 must be with a Z, as
. Therefore, cards with the letter Z were lost.
SLIDE 27
The word “business” is spelled out with 8 letter cards.
B U S I N E S S
One card is taken at random. Which diagram is correct?
The vowels are A, E, I, O, U. There are two half-vowels, W and Y, and the rest are consonants.
SLIDE 28
The word “business” is spelled out with 8 letter cards.
B U S I N E S S
One card is taken at random. There are three vowels, U, I, E; the rest are consonants; 8 letters in all.
The correction option is D.
SLIDE 29
A coin is thrown 60 times.
It lands on heads 45 times.
Which of the following is a possible conclusion?
A As the relative frequency is 45%, we expect the coin to land on heads 45% of the time.
B If the coin is biased, we expect the coin to land on heads exactly 30 times. As 45 is greater than 30, there is evidence that the coin is unbiased.
C The relative frequency is .
D Although the relative frequency is 0.5, we can be sure that the coin is a fair coin.
SLIDE 30
A coin is thrown 60 times. It lands on heads 45 times.
The correct option is
C The relative frequency is .
SLIDE 31
An ordinary fair die is thrown twice. What is the probability of obtaining a 6 twice?
SLIDE 32
On each occasion the probability of getting a 6 is .
The probability of obtaining two 6s is
SLIDE 33
A fair coin is tossed twice. What are all the possible outcomes? If the order in which the coins are tossed does not matter, what is the probability of obtaining (a) no heads, (b) 1 head, (c) 2 heads?
SLIDE 34
A fair coin is tossed twice. What are all the possible outcomes? If the order in which the coins are tossed does not matter, what is the probability of obtaining (a) no heads, (b) 1 head, (c) 2 heads?
Solution
First toss Second toss No. of heads
H H 2
H T 1
T H 1
T T 0
SLIDE 35
? A fair coin is tossed three times. List all the possible outcomes. The first is done for you.
H H H
? What is the probability of obtaining (a) no tails, (b) 1 tail, (c) 2 tails, (d) 3 tails?
SLIDE 36
? A fair coin is tossed three times.
H H H T H H
H H T T H T
H T H T T H
H T T T T T
? What is the probability of obtaining (a) no tails, (b) 1 tail, (c) 2 tails, (d) 3 tails?
SLIDE 37
Two dice are rolled, and the scores are added together.
Complete the table showing the total score.
Number of first die
Number of second die 1 2 3 4 5 6
1 2 3 4
2 3 4
3 4
4
5
6
SLIDE 38
Number of first die
Number of second die 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
Use the table to find the probability when rolling two dice of obtaining a total score of (a) 7, (b) 5, (c) 12
SLIDE 39
Number of first die
Number of second die 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
There are 36 possible outcomes.
(a) Probability of a 7.
Six outcomes have a total of 7.
(b)
(b)
SLIDE 40
Two cards of a pack of 52 playing cards are dealt. What is the probability of the following two cards in exactly that order?
SLIDE 41
There are 52 cards in a pack of playing cards, and there is only 1 Ace of Spades and one King of Spades.
When the first card is dealt there are 52 cards in the pack, so the probability that the first is an Ace of Spades is . But when the second card is dealt, there remain only 51 cards in the pack, so the probability that the second is a King of Spades is .
Probability of an Ace of Spades followed by a King of Spades is
SLIDE 42
Two cards of a pack of 52 playing cards are dealt. What is the probability of the following two cards in either order?
SLIDE 43
Two cards of a pack of 52 playing cards are dealt. What is the probability of the following two cards in either order?
Probability of a Queen of Diamonds followed by a Jack of Diamonds in that order is
But we can have the Jack followed by the Queen. So, the probability is
SLIDE 44
Three cards of a pack of 52 playing cards are dealt.
? Find all possibilities of obtaining the following three cards in any order.
? What is the probability of obtaining these three cards if the order in which they are dealt does not matter?
SLIDE 45
Three cards of a pack of 52 playing cards are dealt.
? There are six combinations of these three cards
? The probability of any one of these hands is
But there are six combinations.
SLIDE 46
There are four aces in a pack of 52 playing cards.
Two cards are dealt.
What is the probability of these two cards being dealt two aces?
SLIDE 47
There are four aces in a pack of 52 playing cards. Two cards are dealt. What is the probability of these two cards being dealt two aces?
Solution
There are four aces in the pack, so the probability of being dealt the first ace is . When one card has been dealt, 51 cards remain, of which three are aces. The probability of the second card being an ace is .
Probability
SLIDE 48
? List all the possible ways in which you can obtain two aces in any order when two cards are dealt from a pack of 52 playing cards.
? How many possibilities are there when two cards are dealt from a pack of 52 playing cards?
?
Use this formula to find the probability of getting two aces in any order.
SLIDE 49
?
12 in all
? There are possibilities are there when two cards are dealt from a pack of 52 playing cards
?
SLIDE 50
Two four-sided dice are thrown.
Dice A has the numbers 1, 2, 3 and 5. Dice B has the numbers 2, 3, 5 and 7.
The scores are added together, and a table is made of the possible scores.
Dice B
2 3 5 7
Dice A 1 3
2 5
3
5
Complete the table and find the probability that the score is an even number.
SLIDE 51
Dice B
2 3 5 7
Dice A 1 3 4 6 8
2 4 5 7 9
3 5 6 8 10
5 7 8 10 12
There are 16 entries, of which 10 are even numbers.
SLIDE 53
Two four-sided dice are thrown.
Dice A has the numbers 1, 2, 3 and 4.
Dice B has the numbers 2, 3, 4 and 5.
The result of dice A is subtracted from the result of dice B. What is the probability of obtaining a negative score?
SLIDE 54
Dice B
2 3 4 5
Dice A 1 -1 -2 -3 -4
2 0 -1 -2 -3
3 1 0 -1 -2
4 2 1 0 -1
There are 16 entries, of which 10 are negative numbers.
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