In a country, the height in centimetres of women aged 18 to 65 can be modelled by a random variable $X _ { 1 }$ following a normal distribution with mean $\mu _ { 1 } = 165 \mathrm {~cm}$ and standard deviation $\sigma _ { 1 } = 6 \mathrm {~cm}$, and that of men aged 18 to 65 by a random variable $X _ { 2 }$ following a normal distribution with mean $\mu _ { 2 } = 175 \mathrm {~cm}$ and standard deviation $\sigma _ { 2 } = 11 \mathrm {~cm}$.\\
In this exercise all results should be rounded to $10 ^ { - 2 }$.

\begin{enumerate}
  \item What is the probability that a woman chosen at random in this country measures between 1.53 metres and 1.77 metres?
  \item a. Determine the probability that a man chosen at random in this country measures more than 1.70 metres.\\
b. Furthermore, it is known that in this country women represent $52 \%$ of the population of people aged between 18 and 65. A person aged between 18 and 65 is chosen at random. They measure more than $1.70 \mathrm {~m}$.\\
What is the probability that this person is a woman?
\end{enumerate}