At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Experience the ease of finding accurate answers to your questions from a knowledgeable community of professionals. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Of course! Let's solve this step-by-step.
1. Understand the problem:
- We are given that 59% of adults with smartphones use them in meetings or classes.
- We need to find the probability that fewer than 3 out of 14 randomly selected adults use their smartphones in such situations.
2. Defining the variables:
- The probability of an adult using a smartphone in meetings or classes ([tex]\( p \)[/tex]) is 0.59.
- The sample size ([tex]\( n \)[/tex]) is 14.
- We are looking for the probability that fewer than 3 adults use their smartphones ([tex]\( X < 3 \)[/tex]).
3. Using the binomial distribution:
- The probability [tex]\( P(X = k) \)[/tex] where [tex]\( k \)[/tex] is the number of successes (i.e., the number of adults using smartphones) in a binomial distribution is given by:
[tex]\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
- Therefore, to find the probability that fewer than 3 adults use their smartphones, we need the cumulative probability for [tex]\( X = 0, 1,\)[/tex] and [tex]\( 2 \)[/tex].
4. Calculate individual probabilities:
- [tex]\( P(X = 0) \)[/tex]: The probability that none of the 14 adults use their smartphones.
- [tex]\( P(X = 1) \)[/tex]: The probability that exactly 1 out of the 14 adults uses their smartphone.
- [tex]\( P(X = 2) \)[/tex]: The probability that exactly 2 out of the 14 adults use their smartphones.
5. Adding up the probabilities:
- The total probability of fewer than 3 adults using smartphones is the sum:
[tex]\[ P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) \][/tex]
Given the calculated values:
- [tex]\( P(X = 0) = 3.7929 \times 10^{-6} \)[/tex]
- [tex]\( P(X = 1) = 7.6414 \times 10^{-5} \)[/tex]
- [tex]\( P(X = 2) = 0.0007147 \)[/tex]
So, the total probability [tex]\( P(X < 3) \)[/tex] is:
[tex]\[ P(X < 3) = 3.792922719491563 \times 10^{-6} + 7.641351625122024 \times 10^{-5} + 0.0007147459385937311 \][/tex]
Adding these values together:
[tex]\[ P(X < 3) = 0.0007949523775644428 \][/tex]
Therefore, the probability that fewer than 3 out of 14 randomly selected adult smartphone users use their smartphones in meetings or classes is approximately 0.0008 when rounded to four decimal places.
1. Understand the problem:
- We are given that 59% of adults with smartphones use them in meetings or classes.
- We need to find the probability that fewer than 3 out of 14 randomly selected adults use their smartphones in such situations.
2. Defining the variables:
- The probability of an adult using a smartphone in meetings or classes ([tex]\( p \)[/tex]) is 0.59.
- The sample size ([tex]\( n \)[/tex]) is 14.
- We are looking for the probability that fewer than 3 adults use their smartphones ([tex]\( X < 3 \)[/tex]).
3. Using the binomial distribution:
- The probability [tex]\( P(X = k) \)[/tex] where [tex]\( k \)[/tex] is the number of successes (i.e., the number of adults using smartphones) in a binomial distribution is given by:
[tex]\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
- Therefore, to find the probability that fewer than 3 adults use their smartphones, we need the cumulative probability for [tex]\( X = 0, 1,\)[/tex] and [tex]\( 2 \)[/tex].
4. Calculate individual probabilities:
- [tex]\( P(X = 0) \)[/tex]: The probability that none of the 14 adults use their smartphones.
- [tex]\( P(X = 1) \)[/tex]: The probability that exactly 1 out of the 14 adults uses their smartphone.
- [tex]\( P(X = 2) \)[/tex]: The probability that exactly 2 out of the 14 adults use their smartphones.
5. Adding up the probabilities:
- The total probability of fewer than 3 adults using smartphones is the sum:
[tex]\[ P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) \][/tex]
Given the calculated values:
- [tex]\( P(X = 0) = 3.7929 \times 10^{-6} \)[/tex]
- [tex]\( P(X = 1) = 7.6414 \times 10^{-5} \)[/tex]
- [tex]\( P(X = 2) = 0.0007147 \)[/tex]
So, the total probability [tex]\( P(X < 3) \)[/tex] is:
[tex]\[ P(X < 3) = 3.792922719491563 \times 10^{-6} + 7.641351625122024 \times 10^{-5} + 0.0007147459385937311 \][/tex]
Adding these values together:
[tex]\[ P(X < 3) = 0.0007949523775644428 \][/tex]
Therefore, the probability that fewer than 3 out of 14 randomly selected adult smartphone users use their smartphones in meetings or classes is approximately 0.0008 when rounded to four decimal places.
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.