A poll of 20 adults is taken in a large city. The purpose is...

A poll of 20 adults is taken in a large city. The purpose is to determine whether they support banning smoking in restaurants. It is known that approximately $60 \%$ of the population supports the decision. Let $X$ represent the number of respondents in favour of the decision. a) What is the probability that 5 respondents support the decision? b) What is the probability that none of the 20 supports the decision? c) What is the probability that at least 1 respondent supports the decision? d) What is the probability that at least two respondents support the decision? e) Find the mean and standard deviation of the distribution.

A poll of 20 adults is taken in a large city. The purpose is to determine whether they support banning smoking in restaurants. It is known that approximately $60 \%$ of the population supports the decision. Let $X$ represent the number of respondents in favour of the decision.
a) What is the probability that 5 respondents support the decision?
b) What is the probability that none of the 20 supports the decision?
c) What is the probability that at least 1 respondent supports the decision?
d) What is the probability that at least two respondents support the decision?
e) Find the mean and standard deviation of the distribution.

Key Concepts

Binomial Distribution

The binomial distribution is used to model the number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success. It applies to situations where there are two possible outcomes (such as support or not support) and is characterized by the number of trials (n) and the probability of success (p) on each trial.

Binomial Probability Mass Function

The binomial probability mass function (PMF) gives the probability of obtaining exactly k successes in n trials. It is computed using the formula P(X = k) = (n choose k) * (p^k) * ((1-p)^(n-k)). This function is used to determine probabilities for specific counts of successes, such as exactly 5 supporters or no supporters in a poll.

Complement Rule in Probability

The complement rule is a fundamental concept in probability that states that the probability that an event occurs is 1 minus the probability that it does not occur. This rule is particularly useful for calculating probabilities such as 'at least one' by considering the complement event (e.g., none support) and subtracting its probability from 1.

Mean and Standard Deviation of Binomial Distribution

For a binomial distribution, the mean (expected value) is calculated as ? = n * p, and the standard deviation is given by ? = sqrt(n * p * (1-p)). These formulas provide measures of central tendency and dispersion, respectively, and are crucial for understanding the average outcome and variability in the number of successes over a series of trials.

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