Download WorkSHEET 10.2 Normal distribution and probability

Maths Quest Maths A Year 12 for Queensland
WorkSHEET 10.2
Chapter 10 The normal distribution and games of chance WorkSHEET 10.2 1
Normal distribution and probability
Name: ___________________________
1
The mean result for an English test for Year 11
students is 60% and the standard deviation
10%.
(a) Draw a normal distribution curve
marking x-scores equivalent to 1, 2 and 3
standard deviations either side of the
mean
(b)
/50
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(a)
1SD = 10%
2SD = 20%
3SD = 30%
What percentage of students obtained
more than 80% for the test?
(b)
80% is 2SD above mean.
95% of scores lie within 2SD above and
below the mean
So 5% of scores lie below and above 2SD from
the mean. This means that 2.5% of scores lie
above 80%.
2
The mean life of a certain battery is 30 hours.
The standard deviation is 1 hour. Within what
limits will the life of the battery almost
certainly lie?
The life of the battery will almost certainly lie
within 3 standard deviations of the mean.
Minimum life  30  3  1 h
 27 h
Maximum life  30  3  1 h
 33 h
So 99.7% of the batteries would almost
certainly have a life in the range 27 h to 33 h.
Where necessary, use the Standard Normal
Tables on page 481 in your textbook to find
the area under the standard normal curve.
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Maths Quest Maths A Year 12 for Queensland
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Chapter 10 The normal distribution and games of chance WorkSHEET 10.2 2
In a normal distribution, find values for each
of the following.
(a) Pz  1.2
(b)
Pz  1.2
(c)
Pz  1.2
(d)
P 1.2  z  1.2
(a)
Pz  1.2  0.8849
(b)
P z  1.2  1  0.8849
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 0.1151
(c)
P z  1.2   Pz  1.2
 0.1151
(d)
P 1.2  z  1.2  1  2  0.1151
 1  0.2302
 0.7698
4
Results for an end of semester Maths A test
had a mean of 58% and standard deviation of
12%. What is the probability that a student
selected at random received 80%?
6
x  x
z
s
80  58

12
22

12
 1.83
P 80%  Px  80
 Pz  1.83
 0.9664
5
At the TAB, a gambler bets $50 on a horse at
4 to 1. If the horse wins:
(a) what amount does this bet win?
(a)
Amount won 
chances of losing
 bet
chances of winning
4
 $50
1
 $200

(b)
how much does the gambler receive?
(b)
Return to gambler  win  bet
 $200  $50
 $250
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Maths Quest Maths A Year 12 for Queensland
Chapter 10 The normal distribution and games of chance WorkSHEET 10.2 3
6
Convert odds of 2 to 1 on to a probability.
7
In a game of two-up, a spinner tosses 2 coins.
What is the probability of the spinner tossing a
HH then a TH?
The odds 2 to 1 on means 1 : 2.
This represents 1 chance of losing to 2 chances
of winning.
So out of the 3 chances there are 2 of winning.
2
 Pwinning  
3
4
6
PHH   0.25
PTH   0.25  0.25
 0. 5
 PHH then TH   PHH   PTH 
 0.25  0.5
 0.125
8
A roulette wheel has 38 compartments: 18 red,
18 black and 2 green. What is the chance of
the moving ball landing in a compartment
coloured:
(a) red?
(b)
green?
(c)
blue?
(a)
Pred  
(b)
Pgreen  
(c)
Pblue  
18
38
9

19
2
38
1

19
0
38
0
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Maths Quest Maths A Year 12 for Queensland
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Chapter 10 The normal distribution and games of chance WorkSHEET 10.2 4
Lucy loves playing Solitaire on the computer.
From experience she has found that she can
complete the game 5 times in 15 attempts.
How many times would Lucy expect to
complete the game in 75 attempts?
Pcompleting  
5
15
1

3
Expect number of completion s in 75 attempts
5
1
  75
3
 25
10
At a fete, 500 tickets are sold at $1 each. The
prize is valued at $300. What is the expected
gain or loss by a person who buys 1 ticket?
300
499
 1 
500
500
300 499


500 500
 199

500
 0.398
Expected gain 
So a loss of 40c is expected.
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