State which type of probability distribution (uniform, binomial, hypergeometric, none of these) would model each situation.

a) A health inspector is in charge of inspecting 75 restaurants, 15 of which have had health code violations in the past. The inspector randomly selects 10 0f the 75 restaurants for inspection. What is the probability that four of these will have had health code violations?

b) It is estimated that 12% of all restaurants in a city have had health code violations. Ten restaurants are selected at random for inspection. What is the probability that four of these will have had health code violations?

c) For a charity lottery, you picked 1, 2, 3, 5, and 8 from the numbers 1 to 20. Five different winning numbers are selected at random. What is the probability of three of your numbers matching the five winning numbers?

d) For a school fundraising draw, 1000 tickets are sold, each with a number from 0001 to 1000. The winning ticket is drawn from a bin. What is the probability of winning the draw?

If, in a probability distribution, the number of successes is counted, then the distribution

A must be binomial

B must be hypergeometric

C may be either binomial or hypergeometric

D may be neither binomial nor hypergeometric

On a TV game show there are nine squares. five of which have a winning sum of money. The contestant selects four different squares. The probability distribution for the number of squares chosen that contain money is

A uniform

B binomial

C hypergeometric

D none of the above

Identify possible random variables for the following experiments and the values the variables may take:

dealing five cards from a deck

Identify possible random variables for the following experiments and the values the variables may take:

naming four members of a committee selected from five grade 11 and seven grade 12 students

Identify possible random variables for the following experiments and the values the variables may take:

cutting a card from a deck

Identify possible random variables for the following experiments and the values the variables may take:

rolling a die 10 times

Identify possible random variables for the following experiments and the values the variables may take:

testing 20 bottles of ginger ale for quality control

Identify possible random variables for the following experiments and the values the variables may take:

selecting a winning square on a TV game show

A game consists of randomly selecting a number from 1 to 15.

Your favourite number is 13, and you are hoping your number will come up.

a) Is this a uniform or binomial distribution? Explain.

b) Rewrite the situation to convert it to uniform or binomial, as appropriate.

At Bill’s Burger Barn. there is a one in eight chance of winning a free hamburger. Nicolas bought a hamburger every day for five days. hoping to win as many free hamburgers as possible.

a) Is this a binomial or hypergeometric distribution? Explain.

b) Rewrite the situation to convert it to binomial or hypergeometric, as appropriate.

For a random draw. 20 slips of paper containing people's names are placed into a bin. Barb noted that four of the names were her friends. Five names will be selected to win a prize. and Barb is hoping at least one of the prizes goes to a friend.

a) Is this a binomial or hypergeometric distribution? Explain.

b) Rewrite the situation to convert it to binomial or hypergeometric, as appropriate.

c) Calculate the probability of success for Barb in each distribution.

d) Which distribution would make Barb happier? Why?

Compare the expectations for cutting a card from a deck four times and for dealing four cards. Then. explain the results

a) for the number of aces

b) for the number of red cards

c) for the number of hearts

A basket contains 20 slips of paper, each with a different student’s name on it. Eight of the names are boys and 12 are girls. Six different names are selected at random, and those students will win fantastic prizes!

a) Explain why this scenario can be modelled using a hypergeometric distribution.

b) Show a full probability distribution for the number of girls who win prizes.

c) Determine the expected number of girls who win prizes using two methods and confirm that they are equal. If not, explain any differences.

d) Rewrite the situation described above to change it to a binomial distribution.

At a fall fair, players in a ring-toss game are successful 8% of the time.

a) Design a problem that would involve a binomial distribution.

b) Design a problem that would involve a hypergeometric distribution.

c) Design a problem that would involve the uniform distribution.

When the population in a binomial distribution is very large and p is very small, it can be modelled using the Poisson distribution, named for the French mathematician Simeon-Denis Poisson (1781—1840). It uses the formula

\displaystyle P(x) = \frac{e^{-np}(n p)^x}{x!} , where e is an irrational number 2.718 28....An estimated 1.5% of the world’s population has green eyes. If 2000 people were selected at random, use the Poisson distribution to calculate the probability that fewer than 10 have green eyes. Compare the results using a graphing calculator. How close is the approximation?