Analysis and Approaches for IBDP Maths Ebook 1 | Page 293

Front Page
96
Paper 2 Section B – Normal distribution
with binomial distribution
Example
The heights of the trees in a park are normally distributed with a mean of 60 metres and a
standard deviation of 6 metres. Trees are classified as short trees if they are shorter than
52.5 metres.
(a)
A tree is selected at random from the park.
(i)
Find the probability that this tree is a short tree.
(b)
(c)
(ii) Given that this tree is a short tree, find the probability that it is shorter than
50 metres.
[6]
Two trees are selected at random. Find the probability that they are both short
trees.
[2]
80 trees are selected at random.
(i)
(ii)
Find the expected number of these trees that are short trees.
Find the probability that less than 11 of these trees are short trees.
[6]
Solution
21
(a) (i) Let H : Height of a tree in the park
The required probability
�P( H � 52.5)
(M1) for valid approach
� 0.105649839
� 0.106
A1 N2
(ii) P( H �50 | H � 52.5)
(R1) for correct probability
P( H � 50 � H � 52.5)
�
P( H � 52.5)
(A1) for correct formula
P( H � 50)
�
P( H � 52.5)
0.0477903304
�
0.105649839
(A1) for correct values
� 0.4523464571
� 0.452
A1 N3
[6]
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