4. Probability

"Blood TypesA person can have one of four blood types: A, B, AB, or O.

If a person is randomly selected, is the probability they have blood type A equal to 1/4? Why or why not?"

"More Genetics In Problem 29, we learned that for some diseases, such as sickle-cell anemia, an individual will get the disease only if he or she receives both recessive alleles. This is not always the case. For example, Huntington’s disease only requires one dominant gene for an individual to contract the disease. Suppose that a husband and wife, who both have a dominant Huntington’s disease allele (S) and a normal recessive allele (s), decide to have a child.

b. What is the probability that the offspring will not have Huntington’s disease? In other words, what is the probability that the offspring will have genotype ss? Interpret this probability.

"

The word "and" in probability implies that we use the ______ Rule.

Suppose that events E and F are independent, P(E) = 0.3 and P(F) = 0.6. What is the P(E and F)?

Double Jackpot Shawn lives near the border of Illinois and Missouri. One weekend he decides to play \$1 in both state lotteries in hopes of hitting two jackpots. The probability of winning the Missouri Lotto is about 0.00000028357 and the probability of winning the Illinois Lotto is about 0.000000098239.

b. Find the probability that Shawn will win both jackpots.

Christmas Lights

Christmas lights are often designed with a series circuit. This means that when one light burns out the entire string of lights goes black. Suppose that the lights are designed so that the probability a bulb will last 2 years is 0.995. The success or failure of a bulb is independent of the success or failure of other bulbs.

a. What is the probability that in a string of 100 lights all 100 will last 2 years?

Earn More Than Your Parents?

In 1970, 92% of American 30-year-olds earned more than their parents did at age 30 (adjusted for inflation). In 2014, only 51% of American 30-year-olds earned more than their parents did at age 30. Source: Wall Street Journal, December 8, 2016.

b. What is the probability that two randomly selected 30-year-olds in 1970 earned more than their parents at age 30?

Quality Control

Suppose that a company selects two people who work independently inspecting two-by-four timbers. Their job is to identify low-quality timbers. Suppose that the probability that an inspector does not identify a low-quality timber is 0.20.

b. How many inspectors should be hired to keep the probability of not identifying a low-quality timber below 1%?

Reliability

For a parallel structure of identical components, the system can succeed if at least one of the components succeeds. Assume that components fail independently of each other and that each component has a 0.15 probability of failure.

c. How many components would be needed in the structure so that the probability the system will succeed is greater than 0.999999?

Bowling

Suppose that Ralph gets a strike when bowling 30% of the time.

a. What is the probability that Ralph gets two strikes in a row?

Audits and Pet Ownership

According to Internal Revenue Service records, 6.42% of all household tax returns are audited. According to the Humane Society, 39% of all households own a dog. Assuming dog ownership and audits are independent events, what is the probability a randomly selected household is audited and owns a dog?

Betting on Sports

According to a Gallup Poll, about 17% of adult Americans bet on professional sports. Census data indicate that 48.4% of the adult population in the United States is male.

d. How will the information in part (c) affect the probability you computed in part (b)?

A Random Process—The Lady Tasting Tea

Ronald Fisher is considered the father of experimental design. Being of English descent, he was having afternoon tea with a colleague. The colleague’s wife entered the room as Fisher was pouring tea. Fisher offered tea to the lady. She politely accepted and requested milk with her tea. Fisher started to pour milk into the tea cup first, but the lady indicated that she preferred her tea be poured first, then the milk. Fisher did not believe that the lady could tell the difference between “milk first” versus “milk second” tea, but the lady insisted she could tell the difference. Being the consummate scientist, Fisher suggested an experiment in which he randomly put milk into the tea first in some instances, and milk into the tea second in others. It turns out, the lady tasting tea was correct in all eight trials.

c. Explain how a coin could be used to simulate the random process of tasting eight cups of tea.

Rolling a Die What is the probability of obtaining 4 ones in a row when rolling a fair, six-sided die? Interpret this probability.

If P(E) = 0.6 and P(E|F) = 0.34, are events E and F independent?