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
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.
Toss a Coin
Possibility or likelihood of something happening is called probability.
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:
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
The letters of the word
CHAOS are placed in a container. Calculate the following probabilities.
- 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 -
HHH; HTT; HHT; HTH; TTH; THT; TTT; THH
- P(three Heads) = 1/8
- P(at least two heads) = 4/8 = 1/2
- P(two Heads and a Tail) = 3/8
Mutually Exclusive Events
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.
Do you find difficulty to remember this? Think of day and night - they are 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
Independent Events
If one event does not influence another event, they are said to be Independent.
E.g.
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)
E.g. 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
Work out the following questions:
1) 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.
2) Two dice are thrown; find the probability of getting the sum of the two
numbers, a prime number.
3)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.
4)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.
5)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.
6)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.
7)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?
8)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.
9)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.
10)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.