Exam 11: Discrete Probability Distributions

Assuming a parent is heterozygous for a particular gene, the likelihood of passing on the dominant allele is 50%-the same as the likelihood of passing on the recessive allele. Assuming that separate births are independent, what is the probability that exactly five out of the next ten children born receive the recessive allele?

On weekend nights, a large urban hospital has an average of 4.8 emergency arrivals per hour. Let X be the number of arrivals per hour on a weekend night at this hospital. Assume that successive arrivals are random and independent. Which distribution does X follow?

The African tigerfish can jump out of water to catch barn swallows flying low over a lake. A study found that, during summer time, 20 birds were caught per day on average by tigerfish in Schroda Dam Lake. Let X be the number of birds caught in flight by tigerfish in Schroda Dam Lake in a day. Assuming that catches are random and independent, we use a Poisson distribution with a mean of 20 to describe the probability distribution of X. What is the probability that X is exactly 20?

State-wide surveys indicate that 14.5% of toddlers in New York are obese. Let the random variable X be the number of toddlers who are obese in a random sample of 20 toddlers from the state of New York. Rounded to four decimal places, what is the probability P(X ≤ 1)?

The African tigerfish can jump out of water to catch barn swallows flying low over a lake. A study found that, during summer time, 20 birds were caught per day on average by tigerfish in Schroda Dam Lake. Let X be the number of birds caught in flight by tigerfish in Schroda Dam Lake in a day. Assuming that catches are random and independent, we use a Poisson distribution with a mean of 20 to describe the probability distribution of X. The probability P(X < 20) = 0.470. What is the probability P(X ≤ 20)?

Assuming that both parents are heterozygous for a recessive gene, the probability that a single offspring develops the recessive trait is 1/4. Assuming separate births are independent, the probability that exactly three out of four offspring develop the trait is given by

On weekend nights, a large urban hospital has an average of 4.8 emergency arrivals per hour. Let X be the number of arrivals per hour on a weekend night at this hospital. Assume that successive arrivals are random and independent. What is the probability of observing at least one emergency arrival?

According to the National Institute of Allergy and Infectious Diseases, approximately 3.7% of U.S. adults have a food allergy. A large company plans a lunch reception for its 400 employees. Assume that employees are independent. Let the random variable X be the number of company employees who have a food allergy. What are the approximate values for the mean and standard deviation of the distribution of X?

For which of the following counts would a binomial probability model be reasonable?

State-wide surveys indicate that 14.5% of toddlers in New York are obese. Let the random variable X be the number of toddlers who are obese in a random sample of 20 toddlers from the state of New York. Rounded to four decimal places, what is the probability that none of the toddlers in the sample is obese?

The proportion of U.S. births that result in a birth defect is approximately 1/33 according to the Centers for Disease Control and Prevention (CDC). A local hospital randomly selects five births and lets the random variable X count the number not resulting in a defect. Assume the births are independent. What is the probability that two of the five births do not result in defects?

Patients with infections are randomly selected and the number of distinct species belonging to the Spirochaete phylum (responsible for illnesses such as Lyme disease and syphilis) are recorded. Let X count the number of distinct species found. It is speculated that X follows a Poisson distribution with a mean of 0.25. What is the standard deviation of the distribution of X?

Which of the following would be appropriately modeled by a Poisson distribution?

According to the National Institute of Allergy and Infectious Diseases, approximately 3.7% of U.S. adults have a food allergy. A large company plans a lunch reception for its 400 employees. Assume that employees are independent. Let the random variable X be the number of company employees who have a food allergy. Using a Normal approximation, what is the probability that at least 20 of the 400 employees have a food allergy?

People with type O-negative blood are universal donors whose blood can safely be given to anyone. Only 7.2% of the population has O-negative blood. A mobile blood center is visited by 20 donors in the afternoon. Let X denote the number of universal donors among them. What is the probability that X is at least 2?

According to the National Institute of Allergy and Infectious Diseases, approximately 3.7% of U.S. adults have a food allergy. A large company plans a lunch reception for its 400 employees. Assume that employees are independent. Let the random variable X be the number of company employees who have a food allergy. What is a reasonable approximation of the distribution of X?

Which of the following is a condition for the binomial distribution?

On weekend nights, a large urban hospital has an average of 4.8 emergency arrivals per hour. Let X be the number of arrivals per hour on a weekend night at this hospital. Assume that successive arrivals are random and independent. What is the probability of observing exactly one emergency arrival?

Patients with infections are randomly selected and the number of distinct species belonging to the Spirochaete phylum (responsible for illnesses such as Lyme disease and syphilis) are recorded. Let X count the number of distinct species found. It is speculated that X follows a Poisson distribution with a mean of 0.25. What is the probability of observing fewer than three distinct species?

On weekend nights, a large urban hospital has an average of 4.8 emergency arrivals per hour. Let X be the number of arrivals per hour on a weekend night at this hospital. Assume that successive arrivals are random and independent. What is the standard deviation of the distribution of X?