Answer:
a) 0.1138 = 11.38% probability that 14 of them were very confident their major would lead to a good job
b) 0.0483 = 4.83% probability that 10 of them are NOT confident that their major would lead to a good job
Step-by-step explanation:
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
In which
is the number of different combinations of x objects from a set of n elements, given by the following formula.
![C_{n,x} = \frac{n!}{x!(n-x)!}](https://tex.z-dn.net/?f=C_%7Bn%2Cx%7D%20%3D%20%5Cfrac%7Bn%21%7D%7Bx%21%28n-x%29%21%7D)
And p is the probability of X happening.
a. A 2017 poll found that 53% of college students were very confident that their major will lead to a good job. If 30 college students are chosen at random, what's the probability that 14 of them were very confident their major would lead to a good job?
Here, we have that
, and we want to find
. So
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
![P(X = 14) = C_{30,14}.(0.53)^{14}.(0.47)^{16} = 0.1138](https://tex.z-dn.net/?f=P%28X%20%3D%2014%29%20%3D%20C_%7B30%2C14%7D.%280.53%29%5E%7B14%7D.%280.47%29%5E%7B16%7D%20%3D%200.1138)
0.1138 = 11.38% probability that 14 of them were very confident their major would lead to a good job.
b. A 2017 poll found that 53% of college students were very confident that their major will lead to a good job. If 30 college students are chosen at random, what's the probability that 10 of them are NOT confident that their major would lead to a good job?
10 not confident, so 30 - 10 = 20 confident. This is P(X = 20).
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
![P(X = 20) = C_{30,20}.(0.53)^{20}.(0.47)^{10} = 0.0483](https://tex.z-dn.net/?f=P%28X%20%3D%2020%29%20%3D%20C_%7B30%2C20%7D.%280.53%29%5E%7B20%7D.%280.47%29%5E%7B10%7D%20%3D%200.0483)
0.0483 = 4.83% probability that 10 of them are NOT confident that their major would lead to a good job