When an experiment is repeated, the ratio of the number of a certain event and the total number of events is called relative frequency - RF.

RF = number of an event / total number of events

**E.g.**

You toss a coin 50 times and get 24 heads and 26 tails. The relative frequencies are as follows:

RF(Head) = 24 / 50 RF(Tail) = 26 / 50

When the number of repetitions of the experiment is increased, the relative frequency seems to be approaching a fixed value. This value is called **probability.
**

The following programme shows how a large number of repetitions bring the relative frequency closer to the probability, i.e. 0.5.

Head | Tail | Total | RF(Head) | RF(Tail) |

Possibility or likelihood of something happening is called *probability*.

Probability = no of events / total number of events

Toss a coin and you get **Head** and **Tail:**

P(Head) = 1/2 ; P(Tail) = 1/2

Probability can take any value between 0 and 1, inclusive: if the event is certain to happen, the probability is 1; if the event is impossible, the probability is 0.

0 | 0.25 | 0.5 | 0.75 | 1 |

Impossible | Very unlikely | 50 - 50 chance | Very likely | Certain |

Now you can see where the following events lie in the probability chart - just move the mouse over them and match the colour:

- David Beckam plays for England again
- Tony Blair will be Prime Minister of United Kingdom in year 2015
- GCSE students are going to get 14 days of study leave before the exam
- Prince Charles will be the King of England in 10 years time
- In April, we move the clock forward
- The next James Bond will be a blond
- If you become an archaeologist, your career will be in ruins!
- Might Tyson, an ex boxer embarking on a change in career - possibly a race jockey

Outcomes

**E.g.1**

Nilup throws a die. Calculate the following probabilities:

- Getting a prime number
- Getting a number less than or equal 4
- Getting a multiple of three

- P(prime) = 3/6 = 1/2
- P(number ≤ 4) = 4/6 = 2/3
- P(multiple of 3) = 2/6 = 1/3

**E.g.2**

Adam throws a die and toss a coin at the same time. Find the following probabilities:

- Getting a Head and a prime number
- Getting a Tail and a multiple of two
- Getting a tail and an odd number

- P(Head and Prime) = 3/12 = 1/4
- P(Tail and multiple of two) = 3/12 = 1/4
- P(Tail and odd number) = 3/12 = 1/4

**E.g.3**

- Getting a vowel
- Getting a consonant
- Getting a letter from the first half of the alphabet

- P(Vowel) = 2/5
- P(Consonant) = 3/5
- P(Letter from the First Half) = 3/5

**E.g.4**

Three red balls, two green balls and five blue balls are in a container. Calculate the following probabilities:

- Getting a red ball
- Getting a red or blue ball
- Not getting a blue ball

- P(Red) = 3/10
- P(Red or Blue) = 8/10 = 4/5
- P(Not getting a blue) = 5/10 = 1/2

**E.g.5**

A coin is tossed three times. Find the following probabilities:

- Getting three Heads
- Getting at least two heads
- Getting two Heads and a Tail

The space diagram looks like this -

- P(three Heads) = 1/8
- P(at least two heads) = 4/8 = 1/2
- P(two Heads and a Tail) = 3/8

**E.g.6**

Two dice are thrown. Find the following probabilities.

- Getting two equal numbers
- The sum of the two numbers being less than 7
- Getting at least one even number

- 6/36 = 1/6
- 15/36 = 5/12
- 27/36 = 3/4

If two events cannot happen at the same time, they are said to be **mutually exclusive**.

**E.g.**

Someone throws a die; look at the following events:

**Event A - getting a prime number****Event B - getting number 2****Event C - getting an odd number**

A and B can happen at the same time - not mutually exclusive.

A and C can happen at the same time - not mutually exclusive.

B and C cannot happen at the same time - mutually exclusive.

'OR' rule for Mutually Exclusive Events

If Event A and Event B are mutually exclusive,

**P( A or B ) = P(A) + P(B)**

**E.g.**

Gehan throws a die. Calculate the probability of getting number 2 or an odd number.

P(2) = 1/6; P(odd) = 1/2

Since these two events are mutually exclusive,

P(2 or Odd) = 1/6 + 1/2 = 4/6 = 2/3

If one event does not influence another event, they are said to be **Independent.**

You throw a die and toss a coin. The outcome of coin does not affect that of die and vice versa; so, these events are independent.

However, now look at the following events:

- Event A - you go to school by bus in time
- Event B - the bus is late

Event A and Event B are **not independent**; B certainly influences
A.

'AND' rule for Independent Events

If Event A and Event B are independent,

**P( A and B ) = P(A) X P(B)**

Natalie throws a die and toss a coin. Calculate the probability of getting a Head and an even number.

P(Head) = 1/2; P(even) = 1/2

Since these two events are independent,

P(Head and even) = 1/2 x 1/2 = 1/4

**E.g.1**

The following button can create a random number between 1 and 9, when you move mouse over it. Find the probability of getting a multiple of 3. Calculate the probability of getting a number greater than 5 as well. What is the probability of getting any number between 1 and 9?

**E.g.2**

The probability of having black hair, red hair and brown hair are 0.6, 0.2 and 0.1 respectively. Find the following probabilities.

- Having red or brown hair
- Having red or black hair
- Having black or brown hair

**E.g.3**

The probability of Mia getting a smartphone for her forthcoming birthday from her mom is 0.7 and probability of getting the same from her grandmother is 0.2.If the elders have not discussed the birthday of Mia, what is the probability of getting a phone from both? Justify the answer.

**E.g.4**

Beatrice has noticed that the electric kettle she buys from a certain catalogue does not last more than a year with malfunctioning could occur at any time during that period. She decided to buy 3 of them in the hope that the probability of all three malfunctioning can be kept to a minimum. If the probability of malfunctioning of a kettle is *x*, show algebraically that she is right.

**E.g.5**

Jason plays for the local football team. The probability of him scoring a goal is ⅔ and getting injured is ⅗;. The probability of his team winning is ¼. Calculate the following probabilities.

- Jason scoring a goal and his team losing
- Jason getting injured and his team winning
- Jason not getting injured, scoring a goal and his team losing

**E.g.6**

The probability of a girl wearing glasses in a certain convent is 0.8 and the probability of a girl being left-handed is 0.6. The probability of a student wearing glasses **and** being left-handed is 0.4. Find the probability of an individual from the convent wearing glasses **or** being left handed. Are the events independent? Explain the answer.

**E.g.7**

The number of people who visit Jack's local gym on Weekdays is 120. At weekends, the number goes up by 40. 20 of them visit both on weekdays and at weekends. Find the probability of his customers visiting on weekdays **or** at weekends.

**E.g.8**

There are 30 students in a class:15 of them do geography; 8 of them do history; 2 of them do neither. Find the probability of a student doing both subjects. Hence or otherwise, find the probability of a student doing geography **or** history.**Hint:** A Venn diagram may help.

**E.g.8**

There are 20 kids in a nursery:12 of them like milk; 4 of like apple juice; 2 of like neither. Find the probability of a kid likes milk **not** apple juice. **Hint:** A Venn diagram may help.

**Work out the following questions:**

- The probability of getting a Head from a biased-coin is 2/3. It is tossed three times. Find the probability of getting three Heads.
- Two dice are thrown; find the probability of getting the sum of the two numbers, a prime number.
- The probability of Maria getting an A in maths is 0.7. The probability of her getting an A in History is 0.4. Find the probability of her getting an A in maths, but not in History.
- Tania will get Play Station-3 from two aunts - one from Australia and the other one from America. However, the two aunts are not on speaking terms. The probability she gets it from aunt in Australia is 0.7 and from aunt in America is 0.4. Calculate the probability of Tania getting the Play Station-3.
- There are 30 students in a class:15 of them do maths;10 do biology; and 5 do both. Find the probability of a students doing both and a student doing neither.
- The letters of the word 'BASE' are placed in a container. Three letters are taken at random. Find the probability of making a 'pronounceable' word.
- The probability of a marksman firing at a target is 0.75. How many times should he fire at it, if he wants to hit it 25 times?
- Tina has two choices to go to work: go by bus and by train: the respective probabilities are 0.8 and 0.5. The probability of being late in each mode is 0.6 and 0.3 respectively. Calculate the probability that Tina is late on arrival for work.
- Two smoke alarms are fitted in a kitchen. The probability of one of them going off is 0.3. Find the probability of just one of them going off in one week.
- The probabilities of three students getting A* for maths are 0.9, 0.7, and 0.4 respectively. Find the probability of none of them getting A*. Calculate the probability of at least two of them getting A* as well.

For a tutorial on **tree diagrams,** please click here.

Maths is challenging; so is finding the right book. K A Stroud, in this book, cleverly managed to make all the major topics crystal clear with plenty of examples; popularity of the book speak for itself - 7^{th} edition in print.