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Stephenson College
HNC Mechanical/Electrical Engineering
Probability
P(certain event) = 1
P(impossible event) = 0
Evens, e.g.Toss a coin:
P(Head) = 0.5
(or ½ or 50%)
Complementary Probabilities
If P(A) = p, then P(A does not occur)= 1 – p
E.g. If probability of “success” = 0.7, then probability of “failure” = 1 – 0.7 = 0.3
Expectation
(i.e.
average occurrence) = np
where n = number of trials, p = probability of success
Example:
throwing a dice
p = probability of throwing a six = 1/6
If you throw a dice 60 times, n = 60, and the expectation of a six = 60 × 1/6 = 10
i.e. you would expect to get 10 sixes out of 60 throws.
Some Definitions
Independent events are those whose probabilities are unaffected by the previous event.
(E.g. Tossing a coin twice. The result of the second toss is not affected by the result of
the first toss).
Dependent events have probabilities which are altered by the result of the previous
event.
(E.g. Sampling from a batch of components when there is no replacement)
Mutually exclusive events cannot both occur at the same time, i.e. there is no overlap.
(E.g. Picking a Heart or a Club from a pack of cards are mutually exclusive since a card
cannot be both a Heart and a Club at the same time.
Hearts
Clubs
However picking an ace and a Club are not mutually exclusive since there is a card
which is both an ace and a Club).
Aces
Ace
of
clubs
H.M.Lorimer
Clubs
Page 1
Numerical methods
Stephenson College
HNC Mechanical/Electrical Engineering
Overlap between A and B
P ( A or B) = P(A) + P(B) – P(A  B)
If A and B are mutually exclusive then P(A  B) = 0 and P ( A or B) = P(A) + P(B).
Example
P(Heart or Club) = P(Heart) + P(Club) – P(Both Heart and Club)
= P(Heart) + P(Club) – 0
=¼+¼
=½
P (Ace or Club) = P(Ace) + P(Club) – P( Ace of Clubs)
= 4/52 + 13/52 – 1/52
= 16/52 = 0.3077
First, then events (one event AND another)
P ( A and B) = P(A) × P(B)
i.e. The probability of event A then event B happening = P(A) × P(B)
Example
P(picking an Ace, then another Ace) = 4/52  3/51 = 0.00452
(assuming the card is NOT replaced)
Answers to questions on page 3
1(a) 0.6
(b) 0.2
(c) 0.15
( P(neither satisfactory) = 0.05)
2 (a) 0.64
(b) 0.32
( P (neither have a phone = 0.04)
3 For any packet P(packet has less than 20 pins) = 0.04
P( one packet less than 20, other has 20 or more) = 0.04  0.96 + 0.96  0.04 =
0.0768
4 (a) 0.3403 (b) 0.0117
(c) 0.3597
(d) 0.1264
(a) 0.3382 (b) 0.0109
(c) 0.3627
(d) 0.1275
5 (a) 0.00714 (b) 0.01786 (c) 0.125
(d) 0.1964
H.M.Lorimer
Page 2
Numerical methods
Stephenson College
HNC Mechanical/Electrical Engineering
Questions
Example 1:
The probability that a component A will operate satisfactorily for 5 years is 0.8 and that B
will operate satisfactorily over that same period is 0.75.
Find the probabilities that in a 5 year period:
(a) both components operate satisfactorily
(b) only component A will operate satisfactorily, and
( c) only component B will operate satisfactorily,
Example 2:
In a particular street, 80% of the houses have telephones. If two houses selected at
random are visited, calculate the probabilities that
(a) they both have a telephone
(b) one has a telephone but the other does not.
Example 3:
Veroboard pins are packed in packets of 20 by a machine. In a thousand packets, 40
have less than 20 pins. Find the probability that if 2 packets are chosen at random, one
will contain less than 20 pins and the other will contain 20 pins or more.
Example 4:
In a box containing 120 transistors 70 are satisfactory, 37 give too high a gain under
normal operating conditions and the remainder give too low a gain. Calculate the
probability that when drawing two transistors in turn at random, with replacement, of
having
(a) two satisfactory
(b) none with low gain
(c) one with high gain and one satisfactory
(d) one with low gain and one satisfactory
Determine the probabilities in (a), (b) and (c) above if the transistors are drawn without
replacement.
Example 5:
The probabilities of an engine failing are given by: p1, failure due to overheating; p2,
failure due to ignition problems; p3, failure due to fuel blockage. When p1 = 1/8, p2 = 1/5,
and p3 = 2/7, determine the probabilities of:
(a) all three failures occurring
(b) the first and second but not the third failure occurring
(c) only the second failure occurring
(d) the first or the second failure occurring but not the third.
H.M.Lorimer
Page 3
Numerical methods
Stephenson College
HNC Mechanical/Electrical Engineering
Practice Questions
1.
When an ordinary, fair die is thrown, what is the probability of obtaining:
i.
5
ii.
1 or 2?
iii.
a number other than 3?
iv.
What is meant by a ‘fair’ die?
2.
Alan has an ordinary pack of 52 playing cards. He takes a card at random. What
is the probability that it is
i.
a 9?
ii.
a club?
iv.
a black 8?
v.
not a diamond?
3.
A tennis club’s membership consists of 27 males (including 5 juniors) and 23
females (including 6 juniors). The Club receives a ticket for the Wimbledon
Championships and allocates it to a member chosen at random. What is the
probability that the chosen member is
i.
a male?
ii.
a female junior?
iii.
not a junior?
4.
Danielle has a battery-driven clock on her mantelpiece. What is the probability that
the battery will run out when the clock is showing a time
i.
between six o’clock and nine o’clock?
ii.
between two o’clock and quarter to three?
5.
State whether you agree or disagree with each of the following statements. Give
reasons for your answer.
i. ‘The probability that a person, chosen at random, was born on a Sunday, is
1
’.
7
ii. ‘Silver Knight is one of twenty horses entered for the Grand National. The
probability of Silver Knight winning it is
1
’.
20
iii. ‘There are four candidates at an election. The probability of each candidate
winning the election is
1
’.
4
1
iv ‘The probability that the sun will shine in London on July 15th is ’.
2
H.M.Lorimer
Page 4
Numerical methods
Stephenson College
6.
HNC Mechanical/Electrical Engineering
Mary owns a gift shop and each week she receives a consignment of 50
ornaments. Over the last 20 weeks she has recorded the number of breakages
per consignment as follows
Number of Breakages
0
1
2
3
4 or more
Number of Consignments
10
4
2
1
3
On the basis of these data estimate the probability that the number of breakages in
the next consignment will be
i.
zero
ii.
1
iii.
2 or more
7.
In the last 365 days a dispensing machine has been out of action for a total of 292
hours. This was due to short-term faults which occurred at random. Taking these
figures to be typical, estimate the probability that when Paul approaches the
machine to make a purchase, he will get served.
8.
Kaniz records her journey time to work as follows
Time (minutes)
40 to 50
50 to 60
60 to 70
70 to 80
i.
ii.
iii.
9.
Number of Journeys
16
14
13
7
How many days do her records cover?
On the basis of the information collected, estimate the probability that her
next journey to work will take
a)
more than fifty minutes;
b)
less than an hour.
Discuss why these might not be good predictions.
Sara and Nick go to the sports centre twice a week to play badminton, but never
book ahead. If no court is available they go swimming instead. In the last 10
weeks they have played badminton 4 times on Tuesdays and 7 times on
Saturdays. On Friday night Nick tells Sara that there is a 55% chance that they
will play badminton tomorrow. Explain how Nick comes to this conclusion. Is he
justified?
H.M.Lorimer
Page 5
Numerical methods
Stephenson College
HNC Mechanical/Electrical Engineering
10. Ashby Ladies’ hockey team plays a league match on Saturday against the team
at the top of the league.
(a) What are the possible outcomes of the game?
(b) Are these outcomes equally likely?
11. Which of these situations are independent events? Explain your answers
a. Tossing a coin twice
b. Picking (and eating) two Smarties from a bag of 10 Smarties of mixed
colours. Are the colours picked independent?
c. Passing an exam in Maths and an exam in Science
d. Rolling two dice and getting a six on both.
12. On your way to college you encounter two sets of traffic lights. The probability
that the first set is green is 0.4. If the first is green then the probability that the
second set is green is 0.8, while if the first is red then the probability that the
second is green is 0.6.
Draw a tree diagram
Calculate the probability that you have to stop at:
(a) both sets of lights
(b) neither set of lights
13. A spinner has a probability of ½ of landing on red and equal probabilities of ¼ of
landing on green or yellow. The spinner is spun twice.
a. Draw a tree diagram to illustrate all the possible outcomes of the two
spins.
b. Calculate the probability of getting 2 yellows
c. Calculate the probability of getting the same colour both times.
d. What is the probability of getting a different colour on the two spins?
e. What is the probability of getting green on neither of the two spins?
H.M.Lorimer
Page 6
Numerical methods
Stephenson College
HNC Mechanical/Electrical Engineering
Answers:
10 (a) win, lose, draw
(b) no
11 (a) independent, one result does not affect the next
(b) no, because if you eat a red Smartie there will be one fewer red to pick next time
(c) Possibly: do you believe that people who are good at maths are also good at
science
(d) independent
12.
first set
second set
0.8
0.4
Green
GG
0.32
0.2
Red
GR
0.08
0.6
Green
RG
0.36
Red
RR
0.24
Green
Red
0.6
0.4
(a)
(b)
P(stop at both sets) = 0.24
P(stop at neither set) = 0.32
13.
First spin
Red
½
second spin
½
Red
RR
¼
¼
green
RG
1/8
¼
Yellow
RY
1/8
½
Red
GR
1/8
¼
green
GG
1/16
¼
Yellow
GY
1/16
½
Red
YR
1/8
¼
green
YG
1/16
Yellow
YY
1/16
¼
Green
¼
Yellow
¼
H.M.Lorimer
Page 7
a) tree
b) P(y,y) = 1/16
c) P(same colour)=
P(RR or GG or YY) =
¼ + 1/16 + 1/16 = 6/16
or 3/8
d) P(different colour) =
1 – 3/8 = 5/8
e) P(G on neither) =
¼ + 1/8 + 1/8 + 1/16 =
9/16
Numerical methods