An athlete believes that her times for running 200 metres in races are normally distributed with a mean of 22.8 seconds.
Given that her time is over 23.3 seconds in \(20 \%\) of her races, calculate the variance of her times.
The record over this distance for women at her club is 21.82 seconds. According to her model, what is the chance that she will beat this record in her next race?
(3 marks)
The events \(A\) and \(B\) are such that
$$\mathrm { P } ( A ) = \frac { 5 } { 16 } , \mathrm { P } ( B ) = \frac { 1 } { 2 } \text { and } \mathrm { P } ( A \mid B ) = \frac { 1 } { 4 }$$
Find
\(\mathrm { P } ( A \cap B )\),
\(\mathrm { P } \left( B ^ { \prime } \mid A \right)\),
\(\mathrm { P } \left( A ^ { \prime } \cup B \right)\),
Determine, with a reason, whether or not the events \(A\) and \(B\) are independent.