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A large cable company reports the following:
80% of its customers subscribe to its cable TV service.
44% of its customers subscribe to its Internet service.
29% of its customers subscribe to its telephone service.
25% of its customers subscribe to both its cable TV and Internet service.
20% of its customers subscribe to both its cable TV and phone service.
23% of its customers subscribe to both its Internet and phone service.
15% of its customers subscribe to all three services.
Consider the chance experiment that consists of selecting one of the cable company customers at random. Calculate the following probabilities.
A) P(cable TV only).
B) P(Internet | cable TV).
C) P(exactly services).
D) P(Internet and cable TV only).

Sagot :

MrRoyal

Answer:

[tex]P(Cable\ TV\ only) = 50\%[/tex]

[tex]P(Internet\ |\ cable\ TV) = 31.25\%[/tex]

[tex]P(exactly\ 2\ services) = 23\%[/tex]

[tex]P(Internet\ and\ cable\ TV \only) = 23\%[/tex]

Step-by-step explanation:

Given

[tex]Cable\ TV = 80\%[/tex]

[tex]Internet = 44\%[/tex]

[tex]Telephone = 29\%[/tex]

[tex]Cable\ TV\ and\ Internet = 25\%[/tex]

[tex]Cable\ TV\ and\ Telephone = 20\%[/tex]

[tex]Internet\ and\ Telephone = 23\%[/tex]

[tex]All\ Services = 15\%[/tex]

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

First, we calculate n(cable TV only)

This is calculated as:

[tex]n(cable\ TV\ only) = (Cable\ TV) - (Cable\ TV\ and\ Internet) - (Cable\ TV\ and\ Telephone) + (All\ Services)[/tex]

[tex]n(cable\ TV\ only) = 80\% - 25\% - 20\% + 15\%[/tex]

[tex]n(cable\ TV\ only) = 50\%[/tex]

The probability is:

[tex]P(Cable\ TV\ only) = \frac{n(Cable\ TV\ only)}{100\%}[/tex]

[tex]P(Cable\ TV\ only) = \frac{50\%}{100\%}[/tex]

[tex]P(Cable\ TV\ only) = 50\%[/tex]

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

This is calculated as:

[tex]P(Internet\ |\ cable\ TV) = \frac{Cable\ TV\ and\ Internet}{Cable\ TV}[/tex]

[tex]P(Internet\ |\ cable\ TV) = \frac{25\%}{80\%}[/tex]

[tex]P(Internet\ |\ cable\ TV) = \frac{25}{80}[/tex]

[tex]P(Internet\ |\ cable\ TV) = 31.25\%[/tex]

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

This is calculated as:

[tex]P(exactly\ 2\ services) = (Cable\ TV\ and\ Internet - All) + (Cable\ TV\ and\ Telephone - All) + (Internet\ and\ Telephone - All)[/tex]

[tex]P(exactly\ 2\ services) = (25\% - 15\%) + (20\% - 15\%) + (23\%-15\%)[/tex]

[tex]P(exactly\ 2\ services) = (10\%) + (5\%) + (8\%)[/tex]

[tex]P(exactly\ 2\ services) = 23\%[/tex]

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

This is calculated as:

[tex]P(Internet\ and\ cable\ TV \only) = \frac{(Internet\ and\ cable\ TV \only)}{100\%}[/tex]

[tex]P(Internet\ and\ cable\ TV \only) = \frac{23\%}{100\%}[/tex]

[tex]P(Internet\ and\ cable\ TV \only) = \frac{23\%}{1}[/tex]

[tex]P(Internet\ and\ cable\ TV \only) = 23\%[/tex]

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