About 12% of the population of a large country is hopelessly romantic. If two people are randomly selected, what is the probability both are hopelessly romantic/ what is the probability least one is hopelessly romantic?

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21-10-2018
Mathematics

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About 12% of the population of a large country is hopelessly romantic. If two people are randomly selected, what is the probability both are hopelessly romantic/ what is the probability least one is hopelessly romantic?

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JeanaShupp JeanaShupp · Answer 1 · 2019-07-26T08:11:52+02:00

Answer: If two people are randomly selected, then the probability both are hopelessly romantic=0.0144

If two people are randomly selected, then the probability at least one is hopelessly romantic =0.2256

Step-by-step explanation:

Binomial probability distribution formula :-

[tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex], where P(x) is the probability of success in x trials , n is the total number of trials and p is the probability of success.

Given : The probability of population of a large country is hopelessly romantic =0.12

If two people are randomly selected, then the probability both are hopelessly romantic :-

[tex]P(2)=^2C_2(0.12)^2(1-0.12)^{2-2}\\\\=(1)(0.0144)(1)=0.0144[/tex]

If two people are randomly selected, then the probability at least one is hopelessly romantic:-

[tex]P(x\leq1)=P(1)+P(2)=^2C_1(0.12)^1(1-0.12)^{2-1}+0.0144\\\\=(2)(0.12)(0.88)+0.0144=0.2112+0.0144=0.2256[/tex]

MrRoyal MrRoyal · Answer 2 · 2021-10-26T13:40:44+02:00

Probabilities are used to determine the chances of an event

The probability that both are hopelessly romantic is 0.0144
The probability that at least 1 is hopelessly romantic is 0.2256

The given parameters are:

[tex]\mathbf{p = 12\%}[/tex] --- proportion that are hopelessly romantic

[tex]\mathbf{n = 2}[/tex] --- the selected sample

(a) The probability that both are hopelessly romantic

This is calculated as:

[tex]\mathbf{Pr = p^n}[/tex]

So, we have:

[tex]\mathbf{Pr = (12\%)^2}[/tex]

Express as percentage

[tex]\mathbf{Pr = (0.12)^2}[/tex]

[tex]\mathbf{Pr =0.0144}[/tex]

Hence, the probability that both are hopelessly romantic is 0.0144

(b) The probability that at least one is hopelessly romantic

First, calculate the probability that none are hopelessly romantic

This is calculated as:

[tex]\mathbf{Pr =(1- p)^n}[/tex]

So, we have:

[tex]\mathbf{Pr =(1- 12\%)^2}[/tex]

Express as percentage

[tex]\mathbf{Pr =(1- 0.12)^2}[/tex]

[tex]\mathbf{Pr =0.88^2}[/tex]

[tex]\mathbf{Pr =0.7744}[/tex]

Using the complement rule, the probability that at least 1 is hopelessly romantic

[tex]\mathbf{P(At\ least\ 1) = 1 - 0.7744}[/tex]

[tex]\mathbf{P(At\ least\ 1) = 0.2256}[/tex]

Hence, the probability that at least 1 is hopelessly romantic is 0.2256