In an orchard, harvested apples are randomly distributed in packets, each containing 7 fruits. The orchard had a total of 25 apples collected, and 10 fruits were considered of excellent quality. Let X be the number of optimal apples allocated to first package, which of the probability distributions would be adequate to describe the random variable X? (a) Geometric (725) (b) Binomial (7,325) (c) Poisson (3) (d) Geometric (325) (e) Hypergeometric (25,10,7)

Question

In an orchard, harvested apples are randomly distributed in packets, each containing 7 fruits.  The orchard had a total of 25 apples collected, and 10 fruits  were considered of excellent quality. Let X be the number of optimal apples allocated to first package, which of the probability distributions would be adequate to describe the random variable X?   (a) Geometric (725)   (b) Binomial (7,325)   (c) Poisson (3)   (d) Geometric (325)   (e) Hypergeometric (25,10,7)

Ruth Phillips · Accepted Answer

Step 1 Random Variable: In probability theory, a random variable is a variable whose values depend on outcomes of a random experiment. It is identified as a measurable function defined on a probability space that maps from a sample space to the real numbers. Discrete Random Variable: A discrete random variable is a variable whose values can be obtained by counting. Though the values can range infinitely, a discrete random variable is countably infinite. The number of students in a class and the number road accidents in a particular city are examples of discreate random variables. Probability Distribution: Let X be a random variable. Then, the probability distribution of X is defined as the statement of all possible values of X together with their corresponding probabilities. Step 2 Here, we have the total number of apples collected, i.e., 25, of which 10 fruits are of excellent quality. So, there are 25−10=15 fruits that are considered not of excellent quality. Also, apples are randomly distributed in packets, each containing 7 fruits, i.e., in the first box, 7 apples are distributed out of 25 apples. X is the number of optimal apples allocated to the first package. Obviously, this selection is without replacement as the same apples cannot be distributed more than once. So, there are 7−x apples allocated in the first package that are not of optimal quality. Therefore, the probability distribution of X will be given by, P(X=x)={(10x)(157−x)(257),x=0,1,2,...n0,otherwise which implies that X∼Hyper geometric(25,10,7) Answer: The correct answer is: Option (e). Since the distribution of X is hypergeometric, it cannot be binomial, Poisson, or geometric. Hence, the other options are incorrect.

In an orchard, harvested apples are randomly distributed in packets, e

Answered question

Answer & Explanation

New Questions in High school statistics