Question

Social networking Web sites in the United Kingdom. In the United States, MySpace and Facebook are considered the two most popular social networking Websites. In the United Kingdom (UK), the competition for social networking is between MySpace and Bebo. According to Nielsen/ Net Ratings (April 2006), 4% of UK citizens visit MySpace, 3% visit Bebo, and 1% visit MySpace and Bebo.

a. Draw a Venn diagram to illustrate social networking sites in the United Kingdom.

b. Find the probability that a UK citizen visits either the MySpace or Bebo social networking site.

c. Use your answer to part b to find the probability that a UK citizen does not visit either social networking site.

Short answer

1. Fig.1 Venn diagram (shown in step 1)
2. 0.06
3. 0.94

Step-by-step solution

Make a Venn diagram to show how social networking sites are used in the United Kingdom.

Venn diagrams are a highly effective graphical approach for representing the sample space S and its subsets. According to net ratings, 4% of individuals frequent MySpace, 3% visit Bebo, and 1% visit MySpace and Bebo. Let A represent the event in which citizens visit MySpace and B represent the event in which citizens visit Bebo.

Therefore,

$\begin{array}{c}\mathbf{P}\mathbf{\left(}\mathbf{C}\mathbf{\right)}\mathbf{=}\mathbf{4}\mathbf{\%}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{4}}{\mathbf{100}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{04}\end{array}$

$\begin{array}{c}\mathbf{P}\mathbf{\left(}\mathbf{B}\mathbf{\right)}\mathbf{=}\mathbf{3}\mathbf{\%}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{3}}{\mathbf{100}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{03}\end{array}$

$\begin{array}{c}\mathbf{P}\mathbf{\left(}\mathbf{C}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{\%}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{1}}{\mathbf{100}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{01}\end{array}$

The Venn diagram below represents the country's use of social networking sites:

Find the probability of either the MySpace or Bebo

The probability that a person will visit either MySpace or Bebo social networking sites is denoted by $\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cup }\mathbf{B}\mathbf{)}$, and it is determined using the additive rule of probability:

$$\begin{gathered} \mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cup }\mathbf{B}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}\mathbf{B}\mathbf{\right)}\mathbf{-}\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cap }\mathbf{B}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{03}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{01} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{06} \end{gathered}$$

Hence, the required probability is 0.06.

Calculate the probability that a UK citizen will not visit each social networking site using your answer to component b

The chance of a citizen not visiting either social networking site is represented by the probability of the occurrence $\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cup }\mathbf{B}\mathbf{)}$

$$\begin{gathered} \mathbf{P}{\left(A\cup B\right)}^{\mathbf{C}}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cup }\mathbf{B}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{06} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{94} \end{gathered}$$

Hence, the required probability is 0.94.