Part A

A competitor participates in an archery competition on a circular target. With each shot, the probability that he hits the target is equal to 0.8.

1. The competitor shoots four arrows. It is considered that the shots are independent. Determine the probability that he hits the target at least three times.
2. How many arrows should the competitor plan to shoot in order to hit the target an average of twelve times?

A general knowledge test consists of a multiple choice questionnaire (MCQ) with twenty questions. For each one, the subject proposes four possible answers, of which only one is correct. For each question, the candidate must necessarily choose a single answer. This person earns one point for each correct answer and loses no points if their answer is wrong.
We consider three candidates:

• Anselme answers completely at random to each of the twenty questions. In other words, for each of the questions, the probability that he answers correctly is equal to $\frac { 1 } { 4 }$;
• Barbara is somewhat better prepared. We consider that for each of the twenty questions, the probability that she answers correctly is $\frac { 1 } { 2 }$;
• Camille does even better: for each of the questions, the probability that she answers correctly is $\frac { 2 } { 3 }$.

1. We denote $X , Y$ and $Z$ the random variables equal to the scores respectively obtained by Anselme, Barbara and Camille. a. What is the probability distribution followed by the random variable $X$? Justify. b. Using a calculator, give the answer rounded to the nearest thousandth of the probability $P ( X \geqslant 10 )$. In the following, we will admit that $P ( Y \geqslant 10 ) \approx 0.588$ and $P ( Z \geqslant 10 ) \approx 0.962$.
2. We randomly choose the copy of one of these three candidates.

We denote $A , B , C$ and $M$ the events:

• $A$: ``the chosen copy is Anselme's'';
• $B$: ``the chosen copy is Barbara's'';
• $C$: ``the chosen copy is Camille's'';
• $M$: ``the chosen copy obtains a score greater than or equal to 10''.

We observe, after correcting it, that the chosen copy obtains a score greater than or equal to 10 out of 20.
What is the probability that it is Barbara's copy? Give the answer rounded to the nearest thousandth of this probability.

When a random variable $X$ follows a binomial distribution $\mathrm { B } \left( 100 , \frac { 1 } { 5 } \right)$, what is the standard deviation of the random variable $3 X - 4$? [3 points]
(1) 12
(2) 15
(3) 18
(4) 21
(5) 24

[Probability and Statistics] A certain math class has 10 groups, each consisting of 3 male students and 2 female students. When 2 people are randomly selected from each group, let $X$ be the random variable representing the number of groups in which only male students are selected. What is the expected value $\mathrm { E } ( X )$ of $X$? (Note: No student belongs to more than one group.) [3 points]
(1) 6
(2) 5
(3) 4
(4) 3
(5) 2

When the trial of simultaneously tossing 2 coins is repeated 10 times, let $X$ be the random variable representing the number of times both coins show heads. Find the variance $\mathrm { V } ( 4 X + 1 )$ of the random variable $4 X + 1$. [3 points]

The random variable $X$ follows a binomial distribution $\mathrm { B } ( 80 , p )$ and $\mathrm { E } ( X ) = 20$. Find the value of $\mathrm { V } ( X )$. [3 points]

The random variable $X$ follows the binomial distribution $\mathrm { B } ( 80 , p )$ and $\mathrm { E } ( X ) = 20$. Find the value of $\mathrm { V } ( X )$. [3 points]

A message is transmitted over a channel where each bit is inverted independently with probability $1-p \in ]0,1[$. Blocks of $r$ bits are transmitted. $X$ denotes the number of inversions during the transmission of a block of $r$ bits. Determine the distribution of $X$, its expectation and its variance.

We consider two urns each containing $A$ balls of which $pA$ are white and $qA$ are black. We draw simultaneously, in an equiprobable manner, $n$ balls from the first urn. We denote $Y$ the number of white balls obtained. We also draw, in an equiprobable manner, $n$ balls from the second urn, but successively and with replacement. We denote $Z$ the number of white balls obtained. What is the distribution of the variable $Z$? Give the expectation and variance of $Z$.

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is:
(1) $\frac { 6 } { 25 }$
(2) 6
(3) 4
(4) $\frac { 12 } { 5 }$

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is:
(1) $\dfrac{12}{5}$
(2) $6$
(3) $4$
(4) $\dfrac{6}{25}$