Question

About 70% of babies born with a certain ailment recover fully.A hospital is caring for six babies born with thiss ailment. The random variable represents the number of babies that recover fully.Decide whether the experiment is a binomial experiment. If it is identify a success, specify the values of n, p, and q, and list the posssible values of the randdom variable x.

Is the experiment a binomial experiment?

A)No

B)yes

What is a success in this experiment?

A) Baby recovers

B) This is not a binomial experiment.

C) Baby doens't recover

Specify the value of n. Select the correct choice below and fill in any answer boxes in your choice.

A) n=

B)this is not a binomial experiment.

specify the value of P. Select the correct choice below and fill in any answer boxes in your choice.

A) p=

B) this is not a binomial exxperiment.

Specify the value of q. Select the correct choice below and fill in any answer boxes in your choice.

A) q=

B) This is not a binomial experiment.

List the possible values of the random variable x.

A) x= 0,1,2,.....6

B)x= 1,2,3.....6

C)x=0,1,2,.....5

D) This is not a binomial experiment.

Due in 20 minutes!!

Answer 1

Concepts and reason

Binomial distribution is one of the important distributions of discrete probability distributions.

The properties of binomial distribution are,

1. The procedure has fixed number of trails.

2. Trails must be independent.

3. Each trail must have two outcomes (success and failure)

4. Probability of success remains constant in all trails.

Combinations gives us in how many ways a particular event can happen or in how many ways one can select a sample from the given population.

Fundamentals

The Binomial probability can be calculated using the formula,

$P\left( {X = x} \right) = \left( \begin{array}{l}\\n\\\\x\\\end{array} \right){p^x}{q^{n - x}}$

$P\left( {X \le x} \right) = \sum\limits_{x = 0}^x {\left( \begin{array}{l}\\n\\\\x\\\end{array} \right){p^x}{q^{n - x}}}$

$P\left( {X < x} \right) = \sum\limits_{x = 0}^{x - 1} {P\left( {X = x - 1} \right)}$

$P\left( {X > x} \right) = 1 - P\left( {X \le x} \right)$

$P\left( {X \ge x} \right) = 1 - P\left( {X < x} \right)$

Where,

p is the probability of success

q is the probability of failure

n is the number of trails

$\left( \begin{array}{l}\\n\\\\x\\\end{array} \right) = \frac{{n!}}{{\left( {n - x} \right)!x!}}$

Where,

n is the size of the set and x is the size of each combination

$n! = n \times \left( {n - 1} \right) \times \left( {n - 2} \right) \times \cdot \cdot \cdot \cdot \times 3 \times 2 \times 1$

(a)

let X denote the number of successes out of a sample of n observations. If each observation is a success with probability p independently of the other observations, then X is a binomial random variable with parameters n and p.

Given $p = 0.70{\rm{ and }}n = 6$

The random variable represents the number of babies that recover fully.

(b)

The random variable represents the number of babies that recover fully.

About 70% of babies born with a certain ailment recover fully.

The success in this experiment is Baby recovers.

(c)

A hospital is caring for six babies born with this ailment.

The sample size is, $n = 6$

(d)

About 70% of babies born with a certain ailment recover fully.

So, the probability of success is, $p = 0.70$

(e)

Total probability is always equal to 1.

$p + q = 1$

From the given information, the probability of success, $p = 0.70$

$\begin{array}{c}\\q = 1 - p\\\\ = 1 - 0.70\\\\ = 0.30\\\end{array}$

(f)

The random variable X represents the number of babies that recover fully.

The hospital is caring for six babies born with this ailment. So, the maximum value for X is 6.

Binomial random variable takes the values from 0 to n

In this experiment, n is 6.

So, x can take the values from 0, 1, 2,…6

Ans: Part a

Yes, it is surely Binomial distribution.