To find the probability that a smartphone chosen at random has either a touch screen or HD movie capability, we can use the formula for the union of two events in probability:
[tex]\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \][/tex]
Where:
- [tex]\( P(A \cup B) \)[/tex] is the probability that a smartphone has either a touch screen or HD movie capability,
- [tex]\( P(A) \)[/tex] is the probability that a smartphone has a touch screen,
- [tex]\( P(B) \)[/tex] is the probability that a smartphone has HD movie capability, and
- [tex]\( P(A \cap B) \)[/tex] is the probability that a smartphone has both a touch screen and HD movie capability.
Given the information:
- [tex]\( P(A) = 0.70 \)[/tex]
- [tex]\( P(B) = 0.55 \)[/tex]
- [tex]\( P(A \cap B) = 0.30 \)[/tex]
We substitute these probabilities into the formula:
[tex]\[ P(A \cup B) = 0.70 + 0.55 - 0.30 \][/tex]
[tex]\[ P(A \cup B) = 0.95 \][/tex]
Now, we need to express this probability as a fraction. We know that 0.95 can be written as the fraction [tex]\(\frac{95}{100}\)[/tex].
To simplify [tex]\(\frac{95}{100}\)[/tex], we can divide both the numerator and the denominator by their greatest common divisor, which is 5:
[tex]\[ \frac{95 \div 5}{100 \div 5} = \frac{19}{20} \][/tex]
Hence, the probability that a smartphone chosen at random has either a touch screen or HD movie capability is [tex]\(\frac{19}{20}\)[/tex].
This fraction matches with option B:
B. [tex]\(\frac{47}{50}\)[/tex]
C. [tex]\(\frac{13}{20}\)[/tex]
D. [tex]\(\frac{43}{53}\)[/tex]
E. [tex]\(\frac{17}{20}\)[/tex]
Therefore, the correct answer is:
E. [tex]\(\frac{17}{20}\)[/tex]
