A die is tossed until the first 6 occurs. What is the probability that it takes 4 or more tosses? Estimate the probability for this geometric distribution by simulating 1000 random samples. Create a histogram of your simulations and describe the shape of the distribution.
UFO sightings have been reported to occur at an average rate of five per hour during certain clear nights. What is the probability that a UFO hunter will spot exactly ten UFOs in two hours?
- Run a random sample of this event and simulate it to estimate the probability and compare it to the exact probability.
- Create a histogram and describe the shape of the distribution.
Let W ∼ Uniform(8, 12). Let M be the growth of a mystical tree in centimeters after being exposed to enchanted unicorn droppings, with the growth rate per day being equal to W.
- Use R to simulate W. Simulate the mean and pdf of M and compare to the exact results.
- Create one graph with both the theoretical density and the simulated distribution.
As an adventurer, you've found a legendary key that can open secret passages in an ancient temple. The key, being centuries old, has a 12% chance of breaking permanently each day. You want to calculate the probability that the key remains intact on each day, from day 1 to day 30. You also want to create a plot of this to demonstrate.
In a thrilling game of "Guess the Jellybeans," there are exactly 100 jellybeans in a jar, with 70 being red and 30 being green. Participants need to draw 5 jellybeans without looking. What is the probability that a participant draws 3 red jellybeans and 2 green jellybeans? Use a simulation to estimate the probability and compare it to the exact probability. Create a histogram of your simulations and describe the shape of the distribution.
Identify the distribution: This is a hypergeometric distribution problem because we have a finite population (100 jellybeans) and we're trying to find the probability of a specific outcome without replacement (3 red and 2 green jellybeans).
Understand the problem: We need to find the probability of witnessing at least 8 students dozing off during a lecture, given that the number of dozing students follows a Poisson distribution with a mean of 5.