Let $n$ be a natural number, $p$ a real number between 0 and 1, and $X_n$ a random variable following a binomial distribution with parameters $n$ and $p$. We denote $F_n = \frac{X_n}{n}$ and $f$ a value taken by $F_n$. We recall that, for $n$ sufficiently large, the interval $\left[p - \frac{1}{\sqrt{n}} ; p + \frac{1}{\sqrt{n}}\right]$ contains the frequency $f$ with probability at least equal to 0.95.
Deduce that the interval $\left[f - \frac{1}{\sqrt{n}} ; f + \frac{1}{\sqrt{n}}\right]$ contains $p$ with probability at least equal to 0.95.

The industrialist claims that only $2\%$ of the valves he manufactures are defective. We assume this claim is true, and we denote $F$ the random variable equal to the frequency of defective valves in a random sample of 400 valves taken from total production.

1. Determine the interval $I$ of asymptotic fluctuation at the $95\%$ threshold of the variable $F$.
2. We choose 400 valves at random from production. We treat this choice as a random draw of 400 valves, with replacement, from production. Among these 400 valves, 10 are defective. In light of this result, can we question, at the $95\%$ threshold, the industrialist's claim?

The following is a table showing customer preference by manufacturer for hiking boots sold at a certain department store.

Manufacturer	A	B	C	D	Total
Preference (\%)	20	28	25	27	100

When 192 customers each purchase one pair of hiking boots, what is the probability that 42 or more customers will choose company C's product, using the standard normal distribution table on the right? [3 points]

$Z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
0.5	0.1915
1.0	0.3413
1.5	0.4332
2.0	0.4772

(1) 0.6915
(2) 0.7745
(3) 0.8256
(4) 0.8332
(5) 0.8413

It is known that $10\%$ of the notebooks displayed in a certain stationery store are products from Company A. When a customer randomly purchases 100 notebooks from this store, find the probability that at least 13 notebooks from Company A are included using the standard normal distribution table below. [3 points]

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
0.75	0.2734
1.00	0.3413
1.25	0.3944
1.50	0.4332

(1) 0.0668
(2) 0.1056
(3) 0.1587
(4) 0.2266
(5) 0.2734

Determine the probability that among 200 randomly selected packages, more than one quarter are returns.

Let $U$ be a binomial random variable with parameters $n \in \mathbf{N}^*$ and $\lambda \in ]0,1[$. Prove the inequality $$d_{VT}\left(p_U, \pi_{n\lambda}\right) \leq n\lambda^2.$$

According to meteorological statistics, in a Nordic city it rains on an average of $45 \%$ of the days. A climatologist analyzes rainfall records from 100 days chosen at random from the last 50 years.
a) (1 point) Express how to calculate exactly the probability that it rained on 40 of them.
b) (1.5 points) Calculate this probability by approximating it using a normal distribution.