5. Binomial Distribution & Discrete Random Variables

National surveys indicate that 36% of people have been in a car accident in the last 5 years. If you randomly sample 10 people, how likely is that exactly 4 have had an accident in the last 5 years?

A biologist is monitoring a large bird sanctuary where a particular bird species is known to have a 70% success rate for each nesting attempt (at least one chick fledges from the nest). This season, she observes 500 independent nesting attempts across the sanctuary.

(B) What is the probability that less than 330 attempts are successful?

For the binomial distribution, which of the following expressions correctly represents the probability of obtaining exactly $k$ successes in $n$ independent trials, each with probability of success $p$?

Consider a binomial experiment with $n=5$ and $p=0.3$. What is the probability that exactly 2 successes occur, that is, compute $P(X=2)$ (to 4 decimals)?

A biologist is monitoring a large bird sanctuary where a particular bird species is known to have a 70% success rate for each nesting attempt (at least one chick fledges from the nest). This season, she observes 500 independent nesting attempts across the sanctuary.

(A) What is the probability that exactly 450 nesting attempts are successful?

Based on historical weather data in a certain city, about 62% of the days are cloudy. Find the mean, standard deviation, and variance for the number of cloudy days in a 30-day month.

A basketball player normally has a 70% chance of making a free throw. The player shoots until finally making a basket, where $x$ is the number of shots they take. Is this a binomial experiment?

You take a 6-question quiz with True/False questions. What is the probability of getting all 6 questions correct by simply guessing?

A biologist is monitoring a large bird sanctuary where a particular bird species is known to have a 70% success rate for each nesting attempt (at least one chick fledges from the nest). This season, she observes 500 independent nesting attempts across the sanctuary.

(C) What is the probability that at least 330 nesting attempts are successful?