Analysis and Approaches for IBDP Maths Ebook 1 | Seite 290

Your Practice Set – Analysis and Approaches for IBDP Mathematics
95
Paper 2 Section B – Conditional probabilities
and independent events
Example
A random variable X is normally distributed with mean, � . The standard deviation of
X is 1.25. It is given that 40% of the data are greater than 20 and less than � .
(a) Find P( X � 20) .
(b) Find the value of � .
The random variable Y is normally distributed with mean � and standard
deviation 1.15. The events X � 20 and Y � 30 are independent, and
P(( X � 20) �( Y � 30)) � 0.45 .
(c) Find � .
(d) Given that Y � 30 , find P( Y � 33) .
[2]
[3]
[5]
[5]
Solution
(a) P( X � 20)
� 0.5 � P(20 � X � �)
(M1) for valid approach
�0.5 � 0.4
� 0.1
A1 N2
(b) P( X �20) � 0.1
� 20 � � �
P�Z
� ��0.1
� 1.25 �
(M1) for standardization
20 � �
�� 1.281551567
1.25
A1
20 � � � (1.25)( � 1.281551567)
� � 21.60193946
� � 21.6
A1 N3
[3]
(c) P(( X � 20) �( Y � 30)) � 0.45
P( X �20) �P( Y �30) � 0.45
(M1) for independent events
(1 � P( X �20)) �P( Y �30) � 0.45
(A1) for correct formula
(1 �0.1) �P( Y �30) � 0.45
(A1) for substitution
P( Y �30) � 0.5
A1
��
� 30
A1 N3
[5]
[2]
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