SPS SPS FM Statistics (SPS FM Statistics) 2021 September

1. a) 5 girls and 3 boys are arranged at random in a straight line. Find the probability that none of the boys is standing next to another boy.
   (3 marks)
   b) A cricket team consisting of six batsmen, four bowlers, and one wicket-keeper is to be selected from a group of 18 cricketers comprising nine batsmen, seven bowlers, and two wicket-keepers.
   How many different teams can be selected?
   (3 marks)
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2. \(\quad \mathrm { P } ( E ) = 0.25 , \mathrm { P } ( F ) = 0.4\) and \(\mathrm { P } ( E \cap F ) = 0.12\)
   a Find \(P \left( E ^ { \prime } \mid F ^ { \prime } \right)\)
   b Explain, showing your working, whether or not \(E\) and \(F\) are statistically independent. Give reasons for your answer.

The event \(G\) has \(\mathrm { P } ( G ) = 0.15\)
The events \(E\) and \(G\) are mutually exclusive and the events \(F\) and \(G\) are independent.
c Draw a Venn diagram to illustrate the events \(E , F\) and \(G\), giving the probabilities for each region.
d Find \(\mathrm { P } \left( [ F \cup G ] ^ { \prime } \right)\)
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3. A group of students were surveyed by a principal and \(\frac { 2 } { 3 }\) were found to always hand in assignments on time. When questioned about their assignments \(\frac { 3 } { 5 }\) said they always start their assignments on the day they are issued and, of those who always start their assignments on the day they are issued, \(\frac { 11 } { 20 }\) hand them in on time.
a Draw a tree diagram to represent this information.
b Find the probability that a randomly selected student:
i always start their assignments on the day they are issued and hand them in on time.
ii does not always hand in assignments on time and does not start their assignments on the day they are issued.
c Determine whether or not always starting assignments on the day they are issued and handing them in on time are statistically independent. Give reasons for your answer.
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4. In a town, \(54 \%\) of the residents are female and \(46 \%\) are male. A random sample of 200 residents is chosen from the town. Using a suitable approximation, find the probability that more than half the sample are female.
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5. The heights of a population of men are normally distributed with mean \(\mu \mathrm { cm }\) and standard deviation \(\sigma \mathrm { cm }\). It is known that \(20 \%\) of the men are taller than 180 cm and \(5 \%\) are shorter than 170 cm .
a Sketch a diagram to show the distribution of heights represented by this information.
b Find the value of \(\mu\) and \(\sigma\).
c Three men are selected at random, find the probability that they are all taller than 175 cm .
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