Question

A large cable company reports the following:

Answer 1

Answer:

$P(Cable\ TV\ only) = 50\%$

$P(Internet\ |\ cable\ TV) = 31.25\%$

$P(exactly\ 2\ services) = 23\%$

$P(Internet\ and\ cable\ TV \only) = 23\%$

Step-by-step explanation:

Given

$Cable\ TV = 80\%$

$Internet = 44\%$

$Telephone = 29\%$

$Cable\ TV\ and\ Internet = 25\%$

$Cable\ TV\ and\ Telephone = 20\%$

$Internet\ and\ Telephone = 23\%$

$All\ Services = 15\%$

Solving (a): A) P(cable TV only).

First, we calculate n(cable TV only)

This is calculated as:

$n(cable\ TV\ only) = (Cable\ TV) - (Cable\ TV\ and\ Internet) - (Cable\ TV\ and\ Telephone) + (All\ Services)$

$n(cable\ TV\ only) = 80\% - 25\% - 20\% + 15\%$

$n(cable\ TV\ only) = 50\%$

The probability is:

$P(Cable\ TV\ only) = \frac{n(Cable\ TV\ only)}{100\%}$

$P(Cable\ TV\ only) = \frac{50\%}{100\%}$

$P(Cable\ TV\ only) = 50\%$

Solving (b): P(Internet | cable TV).

This is calculated as:

$P(Internet\ |\ cable\ TV) = \frac{Cable\ TV\ and\ Internet}{Cable\ TV}$

$P(Internet\ |\ cable\ TV) = \frac{25\%}{80\%}$

$P(Internet\ |\ cable\ TV) = \frac{25}{80}$

$P(Internet\ |\ cable\ TV) = 31.25\%$

Solving (c): P(exactly 2 services).

This is calculated as:

$P(exactly\ 2\ services) = (Cable\ TV\ and\ Internet - All) + (Cable\ TV\ and\ Telephone - All) + (Internet\ and\ Telephone - All)$

$P(exactly\ 2\ services) = (25\% - 15\%) + (20\% - 15\%) + (23\%-15\%)$

$P(exactly\ 2\ services) = (10\%) + (5\%) + (8\%)$

$P(exactly\ 2\ services) = 23\%$

Solving (d): P(Internet and cable TV only).

This is calculated as:

$P(Internet\ and\ cable\ TV \only) = \frac{(Internet\ and\ cable\ TV \only)}{100\%}$

$P(Internet\ and\ cable\ TV \only) = \frac{23\%}{100\%}$

$P(Internet\ and\ cable\ TV \only) = \frac{23\%}{1}$

$P(Internet\ and\ cable\ TV \only) = 23\%$