Binomial distribution

Nowadays more and more teenagers drink alcohol illegally. They might drink beer or stronger drinks. There are cases where teenagers have had alcohol poisoning and have been rushed to the hospital, and there are also cases that teens have lost their lives to alcohol, either directly or indirectly due to a car accident. This phenomenon is becoming worse despite education and efforts by the society to inform the youth about alcohol’s destructive consequences. A recent survey showed the percentages of American students who consume alcohol.
Source: https://www.responsibility.org/get-the-facts/research/statistics/underage-drinking-statistics/According to a survey in 2017 about 12th Grade students in the USA about alcohol consumption;
33% of them reported having had a drink a month before
19% of them had been drunk at least once
17% reported binge drinking
2% reported daily drinking
30 teenagers are randomly selected and are asked whether they have ever been drunk or not. They answer with a “yes” or “no” answer. Therefore, there were only two possible outcomes for each answer, either success or failure. The experiment includes 30 trials where each one is independent of the other trials. As a result, we have n=30 independent trials. The probability of either success or failure is the same, and it is represented by the letter p. The probability that the teenager has been drunk is p=0.19% according to the survey. The probability that a teenager has never been drunk is 0.81. The random variable x represents the number of successful trials and its values are 0,1,2,3,….

29,30.
The expected number of teens who have been drunk at least once is the mean of the binomial distribution. The mean of the distribution is n*p= (30)(0,19)= 5.7. That means that approximately 6 teens out of 30 are expected to have been drunk at least once in their lives. The variance is n*p*(1-p)=30*0.19*0.81=4.617. Using the formula, the standard deviation of the binomial distribution is calculated, and it is found to be equal to (4.617)^1/2=2.15. The probability that exactly 15 out of 30 teenagers have been drunk at least once in their lives is about 0.0001 (0.01%), while the probability that at least 15 out of them have been drunk is about 0.00013(0.013%). The binompdf on the TI 84 was used to find the above numbers.
For the example of the binomial distribution, a 2017 survey among teens was used. The question was if they have ever been drunk and according to the study 19% of teens have been drunk at least once. The random variable x stands for the number of trials among the 30 teens who were asked about whether they have been drunk and responded positively. The probability for every trail is 0,19. Therefore n equals to 30 and P is 0,19.