Part B

Between two phases of the competition, to improve, the competitor works on his lateral precision on another training target. He shoots arrows to try to hit a vertical band, with width 20 cm (shaded in the figure), as close as possible to the central vertical line. The plane containing the vertical band is equipped with a coordinate system: the central line aimed at is the $y$-axis. Let $X$ denote the random variable that, for any arrow shot reaching this plane, associates the abscissa of its point of impact.
It is assumed that the random variable $X$ follows a normal distribution with mean 0 and standard deviation 10.

1. When the arrow reaches the plane, determine the probability that its point of impact is located outside the shaded band.
2. How should the edges of the shaded band be modified so that, when the arrow reaches the plane, its point of impact is located inside the band with a probability equal to 0.6?

The maximum thickness of an avalanche, expressed in centimetres, can be modelled by a random variable $X$ which follows a normal distribution with mean $\mu = 150 \mathrm{~cm}$ and unknown standard deviation. We know that $P ( X \geqslant 200 ) = 0.025$. What is the probability $P ( X \geqslant 100 )$ ? a. We cannot b. 0.025 c. 0.95 d. 0.975 answer because there are missing elements in the problem statement.

Given that in an experiment with 400 plants the value of the random variable $X _ { 400 }$ deviates from the expected value by at most one standard deviation, determine the smallest and largest possible relative frequency of plants that become infested with fungi.

The lifespan of individuals of a certain animal species has a normal distribution with a mean of 8.8 months and a standard deviation of 3 months.\ a) (1 point) What percentage of individuals of this species exceed 10 months? What percentage of individuals have lived between 7 and 10 months?\ b) (1 point) If 4 specimens are randomly selected, what is the probability that at least one does not exceed 10 months of life?\ c) ( 0.5 points) What value of $c$ is such that the interval ( $8.8 - c , 8.8 + c$ ) includes the lifespan (measured in months) of $98 \%$ of the individuals of this species?