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PROBABILITY
Probability is the mathematical measurement of how likely something is to happen
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Every event has probability which lies between 0 and 1 inclusive
The probability of a certainty is 1
The probability of an impossible event is 0
1.1 SAMPLE SPACE, OUTCOMES AND EVENTS
When a football team goes onto a field to play a match, the team knows that one of
three things will happen: The team will win, draw or lose.
Winning, Drawing or losing are outcomes, that is, the different possible results
From a set of outcomes, those that we have special interest in are called events, E
For example if we are interested in the team winning their match, then the event we
are considering is the outcome of ‘winning’.
The sample space, S is the set of all possible outcomes.
Probability is, therefore, the likelihood that an event will happen
Now, if A is an event, then the probability of A
P(A) = n(E)
n(S)
WORKED EXAMPLES
1. When a die is rolled, what is the probability of scoring a prime number?
When a die is rolled, there are six possible outcomes.
S = 1, 2, 3, 4, 5, 6
therefore n(S) = 6
The event we want is to score a prime number, so we are interested in getting the
outcomes of 2, 3 or 5
therefore n(E) = 3
P(scoring a prime number) = n(E) = 3 = 1
n(S)
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2. There are 10 balls in a bag, 6 are blue while 4 are red. If I pick one from the
bag, what is the probability that it is a red ball?
There are 10 balls and therefore 10 possible outcomes:
The event we want is picking a red ball and there are four:
n(S) = 10
n(E) = 4
P(picking a red ball) = n(E) = 4 = 2
n(S) 10 5
MORE EXAMPLES
1. How likely is it that we get a 5 when we roll a dice?
There are a total of 6 possible outcomes so n(S) = 6
The one 5 is the favourable outcome, so
n(E) = 1
P(getting a five) =
2. Find the probability of drawing an ace from a normal pack of cards.
jijijj
There are a total of 52 cards in a pack, so n(S) = 52
The four aces are the favourable outcomes, so n(E) = 4
P(drawing an ace) =
3. A three digit number is formed by picking at random from the numbers
3,7,8. Each number is used only once. Find the probability that the
number formed is even.
Write down the set of all the possible 3-digit numbers formed
What is n(S) =
How many of the members of set are even?
What is n(E) =
Hence find P(the number formed is even) =
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4. A green die and a blue die are thrown together. Both dice have the same
numbers 1 to 6 on their faces. When the dice are thrown, the score is the
sum of the numbers which show on their faces. What is the probability of
getting a score of 9?
SAMPLE SPACE.
Green die
1
2
3
4
5
6
1
1,1
1,2
1,3
1,4
1,5
1,6
Blue die 2
2,1
2,2
2,3
2,4
2,5
2,6
3
3,1
3,2
3,3
3,4
3,5
3,6
4
4,1
4,2
4,3
4,4
4,5
4,6
5
5,1
5,2
5,3
5,4
5,5
5,6
6
6,1
6,2
6,3
6,4
6,5
6,6
What is the total number of possible outcomes
n(S) =
How many of the possible outcomes add up to 9?
N(E)=
P(getting a total score of 9)=
EXERCISE 1.1
1. I Choose a letter from the word MATHEMATICS. What is the probability that
I choose letter T?
2. There are 12 coloured bricks in a sack. 3 are black, 4 are white and 5 are
brown. I choose one of them without looking;
a) What is the probability that I choose a brown brick?
b) What is the probability that I choose a brick which is not white?
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3. A baby could be born on any day of the week. What is the probability that a
certain baby is born on a weekend?
4. Find the probability of drawing a spade from a pack of cards, assuming the
card is selected at random?
5. Annette Babirye writes down each of the 3 digits 4,5,2 in random order.
Find the probability that:
a) the number written down is odd
b) the number written down is greater than 400
1.2.1 MUTUALLY EXCLUSIVE EVENTS
Two events are mutually exclusive if either A or B can happen but not both.
Therefore If A and B are two mutually exclusive events, P(AnB) = 0
Hence P(A or B) = P(AUB) = P(A) + P(B)
WORKED EXAMPLES
1. A bag contains ten identical counters: four green, three red, two blue and
one yellow. A counter is drawn at random. What is the probability that it is
green or blue?
The event ‘it is blue’ and ‘it is green’ are mutually exclusive because a counter
cannot be blue and green at the same time.
P(G or B) = P(GUB) = P(G) + P(B)
= 4/10 +2/10
= 6/10
= 3/5
2. In a lottery six balls are drawn from 49 balls numbered 1 to 49. What is the
probability that the first ball drawn is
a) a multiple of 7
b) a multiple of 8
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c) a multiple of 7 or 8
a) n(S) = 49
Multiples of 7 (7,14,21,28,35,42,49) therefore n(E) = 7
P(multiple of 7) = 7/49
= 1/7
b) n(S) = 49
Multiples of 8 (8,16,24,32,40,48) therefore n(E) = 6
P(multiple of 8) = 6/49
c) P(multiple of 7 or 8) = P(multiple of 7) + P(multiple of 8)
= 1/7 + 6/49
= 13/49
3. In a race, the probability that John wins is 0.3, THE PROBABILITY THAT Paul
wins is 0.2 and the probability that Mark wins is 0.4. Find the probability
that
a) John or Mark wins
b) Neither John nor Paul wins
a) P(John or Mark wins) = P(John wins) + P(Mark wins)
= 0.3 + 0.4
= 0.7
b) P(neither John nor Paul wins) = 1 – P(John or Paul wins)
= 1 – (0.3 + 0.2)
= 0.5
EXERCISE 1.2.1
1. How likely is it that we get a 2 or a 3 when we roll a die?
2. The probability that my wife will wear a blue dress for church next Sunday
is 0.1. The probability that she will wear a white dress is 0.25. What is the
probability that
a) she will wear a blue or white dress
b) she will wear neither a blue dress nor white dress?
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3. From an ordinary pack of 52 playing cards,, the seven of diamonds has
been lost. A card is dealt from a well shuffled pack. Find the probability
that it is a) a diamond b) a queen c) a diamond or a queen
1.2.2 INDEPENDENT EVENTS
Two events A and B are independent if the outcome of A has no effect on the
outcome of B. For example tossing a coin and throwing a die are independent events.
In this case P(A and B) = P(A) x P(B)
That is P(AnB) = P(A) x P(B)
WORKED EXAMPLES
1. Two fair coins A and B are tossed. Find the probability of both showing tails.
The outcome of throwing B is independent of the outcome of throwing A.
For each coin, the probability of a tail is ½ .
i.e P(TA) = ½
also P(TB) = ½
So P(TA TB) = P(TA) x P(TB)
= ½x½
= ¼
This can be shown in a different way
H
T
H
H,H
T,H
T
H,T
T,T
Number of outcomes n(S) = 4
Number of two tails n(E) = 1
P(getting two tails) = ¼
2. Matthew has a choice of three different methods of travelling to work
The probability that he walks, W is ¼
The probability that he uses a bus, B is 1/5
The probability that he uses his car, C is 11/20
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What is the probability that on Monday and Tuesday he:
a) Walks on Monday and uses the bus on Tuesday?
b) Walks once and uses the car once?
a) P(W n B) = P(W) x P(B)
= ¼ X 1/5
= 1/20
b) There are two possible outcomes:
P(Walks on Monday and uses car on Tuesday) or P(Uses car on Monday and
walks on Tuesday)
Therefore P(walks once and uses car once)
= P(WnC) or P(CnW)
= P(WnC) + P(CnW)
= P(W) x P(C) + P(C) x P(W)
= ¼ X 11/20 + 11/20 X ¼
= 22/80
= 11/40
3. The probability that I am late for work is 0.05. Find the probability that on
two consecutive mornings,
a) I am late for work twice
b) I am late for work once
a) P(late twice) = P(late on the first day) and P(late on the second day)
= P(L1) x P(L2)
= 0.05 x 0.05 =
b) Let the probability that I am not late for work will be P(L’)
= P(L1) x P(L2’) or P(L1’) x P(L2)
= 0.05 x 0.95 + 0.95 x 0.05
= 0.0475 + 0.0475
= 0.095
EXERCISE 1.2.2
1. A die is thrown twice. Find the probability of obtaining a number lless
than three on both throws.
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2. A coin is tossed and a die is thrown. What is the probability of obtaining a
head on the coin and an even number on the die?
3. Two men fire at a target. The probability that Alan hits the target is ½ and
the probability that Bob does not hit the target is 1/3. Alan fires at the
target first, then Bob fires at the target. Finnd the probability that
a) Both Alan and Bob hit the target
b) Only one hits the targe
c) Neither hits the target
1.3 Probability Trees (Tree diagrams)
A useful way of tackling probability problems is to draw a probability tree. This
method is illustrated in the following examples.
WORKED EXAMPLE
1. A bag contains four green and six blue marbles which are identical in shape
and mass. A marble is selected at random, and its colour is noted. It is then
returned to the bag and a second marble is selected.
Calculate the probability that the two marbles selected are:
(a) both green
(b) one green and one blue
P(G2)=4/10
P(G1)= 4/10
P(B2)=6/10
P(G2)= 4/10
P(B1) = 6/10
P(B2)=6/10
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(a) P(both green) = P(G) x P(G)
= 4/10 x 4/10
= 16/100
= 4/25
(b) P( one green and one blue) = P(GnB) or P(BnG)
= P(G) x P(B) + P(B) x P(G)
= 4/10 x 6/10 + 6/10 x 4/10
= 24/100 + 24/100
= 48/100
= 12/25
2. In the example above with four green and six blue marbles. If in this case, a
marble is selected at random, its colour noted but It is not returned to the
bag. And then a second marble selected;
Calculate the probability that the two marbles selected are:
(a) both green
(b) one green and one blue
P(G2)=3/9
P(G1)= 4/10
P(B2)=6/9
P(G2)= 4/9
P(B1) = 6/10
P(B2)=5/9
(a) P(both green) = P(G1) x P(G1)
= 4/10 x 3/9
= 12/90
= 2/15
(b) P( one green and one blue) = P(GnB) or P(BnG)
= P(G1) x P(B2) + P(B1) x P(G2)
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= 4/10 x 6/9 + 6/10 x 4/9
= 24/90 + 24/90
= 48/90
= 8/15
EXERCISE 1.2.2
1. A box contains 6 red pens and 3 blue pens. A pen is selected at random
without replacement and the colour is noted. This procedure is performed
a second time then a third time. Find the probability of obtaining
a) red pens
b) 2 red pens and 1 blue pen
1.4 PRACTICE QUESTIONS
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