Question

Some parts of California are particularly earthquake-prone. Suppose that in one metropolitan area, 31% of all home owners are insured against earthquake damage. Four home owners are to be selected at random. Let X denote the number among the four who have earthquake insurance.
(a) Find the probability distribution of X. (Round your answers to. four decimal places.)
(b) What is the most likely value for X?
(c) What is the probability that at least two of the four selected have earthquake insurance? (Round your answer to four decimal places.)

Answer 1

Answer:

a) $P(X=0)=(4C0)(0.31)^0 (1-0.31)^{4-0}=0.2267$  

$P(X=1)=(4C1)(0.31)^1 (1-0.31)^{4-1}=0.4074$  

$P(X=2)=(4C2)(0.31)^2 (1-0.31)^{4-2}=0.2745$  

$P(X=3)=(4C3)(0.31)^3 (1-0.31)^{4-3}=0.0822$  

$P(X=4)=(4C4)(0.31)^4 (1-0.31)^{4-4}=0.0092$  

b) The most likely value is on this case X=1 since have the maximum probability between all the possible values of X

c) $P(X \geq 2) = 1-P(X<2) = 1-P(X\leq 1)= 1-[P(X=0)+P(X=1)]$

$P(X \geq 2) = 1- [0.2267+0.4074]= 0.3659$

Step-by-step explanation:

Previous concepts

A Bernoulli trial is "a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted". And this experiment is a particular case of the binomial experiment.

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

The probability mass function for the Binomial distribution is given as:  

$P(X)=(nCx)(p)^x (1-p)^{n-x}$  

Where (nCx) means combinatory and it's given by this formula:  

$nCx=\frac{n!}{(n-x)! x!}$  

Solution to the problem

Let X the random variable of interest "number among the four who have earthquake insurance"", on this case we can assume that:

$X \sim Binom(n=4, p=0.31)$ with $X=0,1,2,3,4$

Part a

The probability distribution is given by:

$P(X=0)=(4C0)(0.31)^0 (1-0.31)^{4-0}=0.2267$  

$P(X=1)=(4C1)(0.31)^1 (1-0.31)^{4-1}=0.4074$  

$P(X=2)=(4C2)(0.31)^2 (1-0.31)^{4-2}=0.2745$  

$P(X=3)=(4C3)(0.31)^3 (1-0.31)^{4-3}=0.0822$  

$P(X=4)=(4C4)(0.31)^4 (1-0.31)^{4-4}=0.0092$  

Part b

The most likely value is on this case X=1 since have the maximum probability between all the possible values of X

Part c

For this case we want this probability and using complment rule we have:

$P(X \geq 2) = 1-P(X<2) = 1-P(X\leq 1)= 1-[P(X=0)+P(X=1)]$

$P(X \geq 2) = 1- [0.2267+0.4074]= 0.3659$

Answer 2

The probability distribution is as follows:

$P(X=0) = ^4C_0 (0.31)^0 (1-0.31)^4 =0.2267$

$P(X=1) = ^4C_1(0.31)^1 (1-0.31)^3 = 0.4074$

$P(X=3)=^4C_2(0.31)^2(1-0.31)2 = 0.2745$

$P(X=4) = ^4C_3(0.31)^3(1-0.31)^1 = 0.0822$

What is binomial distribution?

binomial distribution with parameters n and p is the discrete probability distribution of the number of successes.

B= ⁿCx * Pˣ * (1 – P)ⁿ⁻ˣ

Where:

B= binomial probability

x = total number of “successes” (pass or fail, heads or tails etc.)

P = probability of a success on an individual trial

n = number of trials.

A) it is given that

n= 4

P=0.31

X=denotes the number among the four who have earthquake insurance.

using the above formulae we can calculate the probability distribution as above, by putting the value of P as 0.31( given).

B) from the distribution we can say that, most likely value of X in this case is X=1 because of having maximum probability.

C)the probability that at least two of the four selected have earthquake insurance = 1 - [P(X=0) + P(X=1)]  = 0.3659

To get more about binomial distribution refer to the link,

https://brainly.com/question/15246027

 

 

 

 

 

Part b

The most likely value is on this case X=1 since have the maximum probability between all the possible values of X

Part c

For this case we want this probability and using complment rule we have: