SPS SPS FM Statistics (SPS FM Statistics) 2021 September

1. 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]
2. 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]

\(P(E) = 0.25\), \(P(F) = 0.4\) and \(P(E \cap F) = 0.12\)

1. Find \(P(E'|F')\) [2 marks]
2. Explain, showing your working, whether or not \(E\) and \(F\) are statistically independent. Give reasons for your answer. [2 marks]

The event \(G\) has \(P(G) = 0.15\) The events \(E\) and \(G\) are mutually exclusive and the events \(F\) and \(G\) are independent.

1. Draw a Venn diagram to illustrate the events \(E\), \(F\) and \(G\), giving the probabilities for each region. [3 marks]
2. Find \(P([F \cup G]')\) [2 marks]

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.

1. Draw a tree diagram to represent this information. [3 marks]
2. Find the probability that a randomly selected student:
   1. always start their assignments on the day they are issued and hand them in on time. [2 marks]
   2. does not always hand in assignments on time and does not start their assignments on the day they are issued. [4 marks]
3. 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. [2 marks]

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. [4 marks]

The heights of a population of men are normally distributed with mean \(\mu\) cm and standard deviation \(\sigma\) cm. It is known that 20% of the men are taller than 180 cm and 5% are shorter than 170 cm.

1. Sketch a diagram to show the distribution of heights represented by this information. [2 marks]
2. Find the value of \(\mu\) and \(\sigma\). [5 marks]
3. Three men are selected at random, find the probability that they are all taller than 175 cm. [2 marks]