According to a study conducted by a university, children who are injured three or more times tend to sustain these injuries during a relatively limited time, usually 1 year or less. If the average number of injuries per year for school-age children is three, what are the probabilities of these events? (Round your answers to three decimal places.)

(1) The probability that a school-age child will sustain 2 injuries in a year is 0.221.

(2) The probability that a school-age child will sustain at least 2 injuries in a year is 0.0804.

(3) The probability that a school-age child will sustain at most 2 injuries in a year is 0.417.

Step-by-step explanation:

Let X = number of injuries per year for school-age children.

The average number of injuries per year for school-age children is, λ = 3.

The random variable X follows a Poisson distribution with parameter λ.

The probability mass function of X is:

(1)

Compute the probability that a school-age child will sustain 2 injuries in a year as follows:

Thus, the probability that a school-age child will sustain 2 injuries in a year is 0.221.

(2)

Compute the probability that a school-age child will sustain at least 2 injuries in a year as follows:

P (X ≥ 2) = 1 - P (X < 2)

             = 1 - P (X = 0) - P (X = 1)

Thus, the probability that a school-age child will sustain at least 2 injuries in a year is 0.0804.

(3)

Compute the probability that a school-age child will sustain at most 2 injuries in a year as follows:

P (X ≤ 2) = P (X = 0) + P (X = 1) + P (X = 2)

Thus, the probability that a school-age child will sustain at most 2 injuries in a year is 0.417.

b.

step-by-step explanation:

the area of a triangle is determined by the formula: base x height * 1/2

you'll need to assign a base to the triangle. based on the vertices given, the simplest base would be the line segment from (-2,1) to (2,1). the distance between any two numbers is found by just subtracting them. here, you should subtract the x values to find the distance of this side.

2 - (-2) = 4.

so the distance (aka the length of the base) is 4. the height of the triangle is the distance between the base and the highest point on the triangle which is (3,4). we should subtract the y values this time to find the height/distance.

4 - 1 = 3

so the height is 3.

base = 4

height = 3

hope this if needed hmu