The number of parking tickets issued in a certain city on any given weekday has a Poisson distribution with parameter μ = 40. (Round your answers to four decimal places.) (a) Calculate the approximate probability that between 35 and 70 tickets are given out on a particular day.

(b) Calculate the approximate probability that the total number of tickets given out during a 5-day week is between 195 and 265.

(c) Use software to obtain the exact probabilities in (a) and (b) and compare to their approximations. Calculate the exact probability that between 35 and 70 tickets are given out on a particular day. Calculate the exact probability that the total number of tickets given out during a 5-day week is between 195 and 265.

1) The hypergeometric probability distribution is closely related to the binomial distribution except that the trials are not independent and the probability of success changes from trial to trial.

True/False

2) The stationarity assumption states that the probability of success in a given binomial distribution does not change from trial to trial.

True/False

3) It is possible for a discrete random variable to assume either a finite number of values or an infinite sequence of values.

True/False

4) Which distribution is used to calculate the probability of a given number of successes for a set number of trials where the only two options are success and failure?

A) The discrete - uniform distribution

B) A bivariate discrete distribution

C) The Poisson distribution

D) The binomial distribution

we will roll a standard six-sided die and try to estimate the probability that each number will come up. If it is a fair die, each number should come up 1/6th of the time.

1. Collecting the data

Put a die in a cup and roll it 30 times. With each roll, record what value came up and list the totals in the table below.
For each number the Relative frequency = # of times the number occurs / # of tosses is your estimate of the probability that the die will land each number.

Given that the die is fair, we would expect each value to come up 30/6 = 5 times. Did that occur?

Enter your results into the table on the board to get a set of frequencies for your first 30 rolls. Repeat the process 3 more times and enter the results in the table below. After each set, make a mental note of how the distribution has changed. After all of the tolls, calculate the overall totals and overall relative frequencies.

2. Analyzing the data and drawing conclusions.
a. Do the frequencies of each number appear to be converging to equal ratios (i.e. is the relative frequency about

1/6th). Using repeated experiments to estimate probability is called Empirical probability. The approach of assuming each value is equally likely is a case of classical probability.

When the number of trials, n, is large, binomial probability tables may not be available. Furthermore, if a computer is not available, hand calculations will be tedious. As an alternative, the Poisson distribution can be used to approximate the binomial distribution when n is large and p is small. Here the mean of the Poisson distribution is taken to be μ = np. That is, when n is large and p is small, we can use the Poisson formula with μ = np to calculate binomial probabilities; we will obtain results close to those we would obtain by using the binomial formula. A common rule is to use this approximation when n / p ≥ 500.

To illustrate this approximation, in the movie Coma, a young female intern at a Boston hospital was very upset when her friend, a young nurse, went into a coma during routine anesthesia at the hospital. Upon investigation, she found that 12 of the last 31,000 healthy patients at the hospital had gone into comas during routine anesthesias. When she confronted the hospital administrator with this fact and the fact that the national average was 5 out of 62,000 healthy patients going into comas during routine anesthesias, the administrator replied that 12 out of 31,000 was still quite small and thus not that unusual.

Note: It turned out that the hospital administrator was part of a conspiracy to sell body parts and was purposely putting healthy adults into comas during routine anesthesias. If the intern had taken a statistics course, she could have avoided a great deal of danger.)

(a) Use the Poisson distribution to approximate the probability that 12 or more of 31,000 healthy patients would slip into comas during routine anesthesias, if in fact the true average at the hospital was 5 in 62,000. Hint: μ = np = 31,000 (5/62,000) = 2.5. (Leave no cell blank. You must enter "0" for the answer to grade correctly. Do not round intermediate calculations. Round final answer to 5 decimal places.)

Shown below are the number of trials and success probability for some Bernoulli trials. Let X denote the total number of successes.

n = 6 and p = 0.3

Determine ​P(x=4​) using the binomial probability formula.

b. Determine ​P(X=4​) using a table of binomial probabilities.

Compare this answer to part​ (a).

Find the probability that the total number of spots showing on 200 fair dice, rolled randomly, is in the range 680 to 720. Give your answer as a percent, without the percent sign. Describe the details of your calculation in the previous problem. Describe the box model (how many tickets in the box model, what numbers are on the tickets, how many of each?). You do not have to write complete sentences. Give the average and SD for the box. Give the expected total number of spots and the SE for the total number of spots. Give the appropriate z value.

A consumer survey has derived the following probability distribution for the number of coffee cups drunk per day

(a) Separately for each gender, what is the probability of drinking more than two cups per day?

Male:

Female:

(b) Calculate for each gender the average number of cups of coffee drunk per day.

Male:

Female

(c) Calculate the probability that, in a group comprising 3 men and 2 women, exactly two are coffee-drinkers.  State any assumptions you are making.

Let n be a random number between 1 and 100000 chosen with uniform probability. Compute

a) The probability that n can be divided by 3

b) The probability that n can be divided by 6

c) The probability that n can be divided by 9

d) The probability that n can be divided by 9 given that in can be divided by 6

e) The probability that n can be divided by 6 given that in can be divided by 9

What is the probability of choosing at random number between 1 and 64 which is 7-Smooth? Explain