The birthday problem considers the probability that two people in a group of a given size have the same birth date. We will assume a 365 day year (no leap year birthdays).

Code set-up

Dobrow 2.28 provides useful R code for simulating the birthday problem. Imagine we want to obtain an empirical estimate of the probability that two people in a class of a given size will have the same birth date. The code

trial = sample(1:365, numstudents, replace=TRUE)

simulates birthdays from a group of numstudents students. So you can assign numstudents or just replace numstudents with the number of students in the class of interest.

If we store the list of birthdays in the variable trial, the code

2 %in% table(trial)

will create a frequency table of birthdays and then determine if there is a match (2 birthdays the same). We can use this code in an if-else statement to record whether a class has at least one pair of students with the same birth date. We then can embed the code within a for-loop to repeat the experiment, store successes in a vector, and then take the average number of successes (a birthday match) across the repeated tasks.

The problems

• Simulate the birthday problem to obtain an empirical estimate of the probability that two people in a class of 23 will have the same birth date. In particular, simulate birthdays for 1000 classes (for(i in 1:1000){...}) each of size 23 and compute the proportion of these classes in which at least one pair of students has the same birth date.

Recall that the true probability is 1-prod(seq(343,365))/(365)^23 which is approximately 50%.

• Using your simulation code, estimate the number of students needed in the class so that the probability of a match is 95%. (You may do this by trial and error.)
• Using your simulation code, find the approximate probability that three people have the same birthday in a class of 50 students.

# [Place code here]

Place your answers to the three items below here:

• [Ans 1]

  The birthday problem considers the probability that two people in a group of a given size have the same birth date. We will assume a 365 day year (no leap year birthdays).

  Code set-up

  Dobrow 2.28 provides useful R code for simulating the birthday problem. Imagine we want to obtain an empirical estimate of the probability that two people in a class of a given size will have the same birth date. The code

  trial = sample(1:365, numstudents, replace=TRUE)

  simulates birthdays from a group of numstudents students. So you can assign numstudents or just replace numstudents with the number of students in the class of interest.

  If we store the list of birthdays in the variable trial, the code

  2 %in% table(trial)

  will create a frequency table of birthdays and then determine if there is a match (2 birthdays the same). We can use this code in an if-else statement to record whether a class has at least one pair of students with the same birth date. We then can embed the code within a for-loop to repeat the experiment, store successes in a vector, and then take the average number of successes (a birthday match) across the repeated tasks.

  The problems

• Simulate the birthday problem to obtain an empirical estimate of the probability that two people in a class of 23 will have the same birth date. In particular, simulate birthdays for 1000 classes (for(i in 1:1000){...}) each of size 23 and compute the proportion of these classes in which at least one pair of students has the same birth date.

• Recall that the true probability is 1-prod(seq(343,365))/(365)^23 which is approximately 50%.

• Using your simulation code, estimate the number of students needed in the class so that the probability of a match is 95%. (You may do this by trial and error.)
• Using your simulation code, find the approximate probability that three people have the same birthday in a class of 50 students.

• # [Place code here]

  Place your answers to the three items below here:

• [Ans 1]

According to a recent Current Population Reports, the population distribution of number of years of education for self-employed individuals in the United States has a mean of 14 and a standard deviation of 3.3. (a) Identify the variable. number of self-employed individuals number of educated individuals in the United States number of self-employed individuals in the United States number of years of education (b) Find the mean and standard deviation of the sampling distribution of for a random sample of size 95. Mean = (1 decimal place) Standard Deviation = (3 decimal places) (c) Find the mean and standard deviation of the sampling distribution of for a random sample of size 424. Mean = (1 decimal place) Standard Deviation = (3 decimal places) (d) Describe the effect of increasing n. The mean increases. The mean stays the same. The mean decreases. The standard deviation of the sampling distribution increases. The standard deviation of the sampling distribution stays the same. The standard deviation of the sampling distribution decreases. (e) If a sample of size 424 is selected from this population, what is the probability that the sample average will be less than 14.2? probability = (3 decimal places)

1, Find the​ Z-scores that separate the middle 59​% of the distribution from the area in the tails of the standard normal distribution..

The​ Z-scores are

2, The number of chocolate chips in an​ 18-ounce bag of chocolate chip cookies is approximately normally distributed with mean 1252 and standard deviation 129 chips.

​(a) What is the probability that a randomly selected bag contains between 1000 and 1400

chocolate​ chips?

​(b) What is the probability that a randomly selected bag contains fewer than 1100 chocolate​ chips?

​(c) What proportion of bags contains more than 1200 chocolate​ chips?

​(d) What is the percentile rank of a bag that contains 1025 chocolate​ chips?

3, The number of chocolate chips in a bag of chocolate chip cookies is approximately normally distributed with mean 1263 and a standard deviation of 118.

​(a) Determine the 26th percentile for the number of chocolate chips in a bag.

​(b) Determine the number of chocolate chips in a bag that make up the middle 97​% of bags.

​(c) What is the interquartile range of the number of chocolate chips in a bag of chocolate chip​ cookies?

The quality control manager of​ Marilyn's Cookies is inspecting a batch of​ chocolate-chip cookies that has just been baked. If the production process is in​ control, the mean number of chip parts per cookie is 6.7. Complete parts​ (a) through​ (d).

A. What is the probability that in any particular cookie being inspected fewer than five chip parts will be found?

B. What is the probability that in any particular cookie being inspected exactly five chip parts will be found?

C. What is the probability that in any particular cookie being inspected five or more chip parts will be found?

D. What is the probability that in any particular cookie being inspected either four or five chip parts will be found?

Luis Suarez is a soccer player with a barely controllable hunger for human flesh. He has, numerous times, bitten an opposing player during a game. If he bites someone again, he will be banned from playing. In any game, the probability he bites an opponent is 0.15. Let the random variable B be the the number of games from now until Luis Suarez bites an opponent, and is subsequently suspended. a) Write out the probability mass function for B. b) What is the probability that Luis bites someone in the fourth game from now? c) If Luis promises not to bite anyone in the next six games, what is the probability that he will last more than 10 games without biting an opponent?

A space shuttle has 6 O-rings. When launched an O-ring has the probability of failure of .014. Whether an O-ring fails is the independent of the other O-rings.

a.)What is the probability that during 23 launches no O-ring fails, but that at least one O-ring will fail during the 24th launch?

b.) What is the probability the space shuttle will go no more than 10 launches without an O-ring failure?

c.) What is the mean number of launches until an O-ring failure?

d.) The first 10 launches of the space shuttle have no O-ring failures. What is the probability the space shuttle will have 15 launches without any O-ring failure?

The label on a bottle of liquid detergent shows the contents to be 12 grams per bottle. The production operation fills the bottle uniformly according to the following probability density function: f(x) = { 8 11:975 < x < 12:100 0 elsewhere where x denotes the grams the bottle will be filled. a) What is the expected value and the standard deviation of X? ( b) What is the probability that a bottle will be filled with between 12 and 12.05 grams? ( c) What is the probability that a bottle will be filled with 12.02 or more grams? ( d) Quality control accepts a bottle that is filled to within 0.02 grams of the number of grams shown on the container label. What is the probability that a bottle of this liquid detergent will fail to meet the quality control standard?

The CDC reports the probability a 25 year old adult will survive to age 35 is 0.986. you select twenty 25-year-old adults at random.

A) What is the probability exactly 19 adults survive to age 35? All 20 survive? At least 19 survive?

B) What is the probability at most 5 adults pass away before 35? at least 15 survive?

C) Find the expected value and explain what it means in context

D) Find the minimum number of 25 year old adults we need to select at random for the probability for the probabilty distribution to be approximatley normal.

E) Under the conditions of part (D) find the upper and lower fences

A dozen eggs contain 12 eggs. A particular dozen is known to have 3 cracked eggs. An inspector randomly chooses 4 eggs from this dozen for inspection. Let X be the number of cracked eggs in the 4 chosen for inspection.
a. Find the probability mass function of X in table form.
b. Find the cumulative distribution function of X in table form.
c. What is the probability that there is at least 1 cracked egg chosen by the inspector?
d. What is the probability that there are at most 2 cracked eggs chosen by the inspector?
e. Find the expected value of X.
f. Find the standard deviation of X.
g. What is the probability that between 0.15 and 2.85 cracked eggs are chosen by the inspector?

Assume that customer arrivals at a barber shop are random and independent of one another, and the number of customer arrivals at a barber shop and the time until the next customer arrives is independent.

(a) In city A, on average, 3 customers arrive at a barber shop every hour. Using an appropriate probability distribution,

(i) find the probability that at least 5 customers arrive at a barber shop every hour.

(ii) A sample of 25 barber shops in city A was obtained. Find the probability that at least 3 barber shops were visited by at least 5 customers.

(iii) A customer has just arrived in a barber shop. Find the probability that the time, until the next customer arrives will be at most 2 hours (from now).