Suppose that the probability that a passenger will miss a flight is 0.0929. Airlines do not like flights with empty​ seats, but it is also not desirable to have overbooked flights because passengers must be​ "bumped" from the flight. Suppose that an airplane has a seating capacity of 52 passengers. ​(a) If 54 tickets are​ sold, what is the probability that 53 or 54 passengers show up for the flight resulting in an overbooked​ flight? ​(b) Suppose that 58 tickets are sold. What is the probability that a passenger will have to be​ "bumped"? ​(c) For a plane with seating capacity of 54 ​passengers, how many tickets may be sold to keep the probability of a passenger being​ "bumped" below 55​%?

The probability that a student correctly answers on the first try (the event A) is P(A) = 0.3. If the student answers incorrectly on the first try, the student is allowed a second try to correctly answer the question (the event B). The probability that the student answers correctly on the second try given that he answered incorrectly on the first try is 0.4. Find the probability that the student answers the question on the first or second try.

a) 0.88

b) 0.12

c) 0.70

d) 0.42

e) 0.58

Two parents are AabbCc and aaBbcC.

What is the probability that their child will have the dominant phenotype for all three alleles shown above. Alleles are independently assorted.

Prove using only the axioms of probability that if A and B are events, then P(A ∪ B) ≤ P(A) + P(B)

Amy’s birthday is on December 6. What is the probability that at least one of the 40 other students has the same birthday as Amy? (Provide a numerical expression, but don’t attempt to simplify. Assume there are 365 days in every year.)

You are conducting a study to see if the probability of catching the flu this year is significantly more than 0.13. You use a significance level of α=0.10α=0.10.

      H0:p=0.13H0:p=0.13
      H1:p>0.13H1:p>0.13

You obtain a sample of size n=692n=692 in which there are 103 successes.

What is the test statistic for this sample?
test statistic = (Report answer accurate to 3 decimal places.)

What is the p-value for this sample?
p-value = (Report answer accurate to 4 decimal places.)

The p-value is...

• less than (or equal to) αα
• greater than αα

This test statistic leads to a decision to...

• reject the null
• accept the null
• fail to reject the null

As such, the final conclusion is that...

• There is sufficient evidence to warrant rejection of the claim that the probability of catching the flu this year is more than 0.13.
• There is not sufficient evidence to warrant rejection of the claim that the probability of catching the flu this year is more than 0.13.
• The sample data support the claim that the probability of catching the flu this year is more than 0.13.
• There is not sufficient sample evidence to support the claim that the probability of catching the flu this year is more than 0.13.

How can probability distribution be used in business management of firms?

What is the probability that every student in a class of 20 will pass the quiz?

-This is a hands-on exercise in probability distributions and sampling. To complete this problem, you will need 25 pennies and a marker. Take 10 pennies and label them 1 through 10, the same number on both sides. If you don’t want to permanently mark your pennies, use a washable marker or put some tape on the 10 marked pennies and write on the tape.

Part A: Human Beings Aren’t Good at Being Random Think about flipping a fair coin 100 times and recording H for heads or T for tails as you go along. You’d end up with 100 random Hs & Ts in a string like:

TTH.....................................................T (first was T, second was T, third was H...100th was T)

Because you have studied statistics you expect roughly 50 Hs—maybe 47 or 51—but probably not 94 Hs and 6 Ts.

I want you to do your best to write out such a random string; you are mentally flipping a coin 100 times in a row. It should take you just a minute or two. Don’t flip any coins, just imagine it. Do this before you read any further.

Now I’d like you to really flip a penny 100 times, recording H or T in order. Well, this would be a pain— you could do this with a penny, but it would waste a lot of your time. We can effectively do the same thing by putting your 10 marked pennies in a jar, shaking them, and pouring them out. Then record, in order, the ten H/Ts that show up. I did this in the picture above, and wrote down THTHHHTHHT because the coins follow the pattern below:

#              1   2     3    4     5 6     7     8     9     10

showing    T   H   T   H   H   H    T    H    H      T

Do this 10 times and write down the resulting 100 H/Ts in order, so you have 100 letters in a row. Now you have two strings of 100 H/Ts each. Email those strings to your instructor. Your instructor will email back to you the rest of Part A after reviewing your strings. (I bet the suspense is killing you).

Part B: Sampling and Random Variable You already have ten marked pennies (ones with numbers from Part A) and 15 unmarked pennies.

Thought experiment: Throw them all in a jar and shake. Without looking, pull three out and record how many of them are marked (have a number). You will get 0, 1, 2, or 3 marked coins.

How many different samples of 3 pennies out of 25 can you get? (Order doesn’t matter.) Answer: 2,300                      Show why 2,300 is the answer.

How many of those samples would have 0, 1, 2, 3 marked coins?

If you draw a simple random sample of size 3, each sample is equally likely. Counting the number of marked coins gives us a discrete random variable, X.

Find the rest of these probabilities. Then find the mean and standard deviation of this discrete random variable.

Now, really and truly put the ten marked coins in a jar with 15 unmarked ones. Shake, pull out three without looking. Write down the number of marked coins. Put them all back, shake, draw again, and count marked coins again. Do this a total of 20 times. Now you have 20 pieces of data. Write down the data set. Compute the sample proportions. How do your sample proportions compare to the probabilities you computed above? Find the mean and standard deviation of the data set. Are the expected value and standard deviation of the random variable close to the mean and standard deviation of the data set? Should they be? Why?

The groundwater concentration of a chemical in a region (X) is a random variable with probability density function ?? (?) = { 1.2(? + ? 2 ) 0 < ? < 1 0 ??ℎ??????

a) Find the cumulative distribution of X.

b) Graphically illustrate the PDF and CDF if

c) Find the probability that X exceeds 0.5.

d) Find the mean, median and mode of X.

e) Find the standard deviation ? and coefficient of variation of X.

f) Find the probability that the concentration is within ±2? of the mean.

g) Determine the quantile estimation function and Graphically illustrate the function.

h) Find the 90%, 95%, and 99% quantiles of X.