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.
