College

Select the correct answer.

A company that manufactures smartphones makes [tex]70\%[/tex] of the phones with a touch screen and [tex]55\%[/tex] of the phones with HD movie capability. [tex]30\%[/tex] of the phones have both a touch screen and HD movie capability.

What is the probability that a smartphone chosen at random has either a touch screen or HD movie capability?

A. [tex]\frac{19}{20}[/tex]
B. [tex]\frac{47}{60}[/tex]
C. [tex]\frac{18}{20}[/tex]
D. [tex]\frac{43}{80}[/tex]
E. [tex]\frac{17}{20}[/tex]

Answer :

To solve this problem, we're looking for the probability that a smartphone chosen at random has either a touch screen or HD movie capability. We are given the following probabilities:

- Probability of a phone with a touch screen, [tex]\( P(T) = 70\% = 0.70 \)[/tex]
- Probability of a phone with HD movie capability, [tex]\( P(H) = 55\% = 0.55 \)[/tex]
- Probability of a phone having both features, [tex]\( P(T \text{ and } H) = 30\% = 0.30 \)[/tex]

To find the probability of a smartphone having either a touch screen or HD movie capability, we use the formula for the union of two events:

[tex]\[ P(T \text{ or } H) = P(T) + P(H) - P(T \text{ and } H) \][/tex]

Substitute the given values into the formula:

[tex]\[ P(T \text{ or } H) = 0.70 + 0.55 - 0.30 \][/tex]

Calculate step-by-step:

1. Add the probabilities of the individual events: [tex]\( 0.70 + 0.55 = 1.25 \)[/tex].
2. Subtract the probability of both events occurring: [tex]\( 1.25 - 0.30 = 0.95 \)[/tex].

Thus, the probability that a smartphone has either a touch screen or HD movie capability is [tex]\( 0.95 \)[/tex].

To express this as a fraction and locate the correct answer from the choices, we convert [tex]\( 0.95 \)[/tex] to a fraction. The equivalent fraction in simplest form is [tex]\(\frac{19}{20}\)[/tex].

Therefore, the correct answer is:

A. [tex]\(\frac{19}{20}\)[/tex]