Midterm II - APMA 3100 - McMillan - Spring 2025

Final Exam drafting

Counting

Suppose a code must be created using 5 letters and 3 numerals. No letter or numeral may be repeated. (There are 26 letters and 10 numerals.) Symbol order matters for codes.

(a) How many unique codes are possible if the letters must come first and the numerals last?

(b) How many unique codes are possible if any ordering pattern of 5 letters and 3 numerals is permitted?

(c) How many ways are there to put 17 guinea pigs into 3 buckets?

Bayes’ Theorem

Urn I has 2 blue and 3 green marbles. Urn II has 2 blue and 5 green marbles.

Toss a biased coin with . If it lands heads, choose Box A; if tails, choose Box B. A single marble is then drawn from the chosen box.

Suppose the experiment is run and a blue marble is drawn.

(a) What is the probability that Box A was chosen?

(b) What is the probability that a second marble drawn from the chosen box will be blue as well?

Joint distributions

Suppose and are discrete RVs with the following matching PMFs:

(a) Find a joint PMF that is compatible with the above and .

(b) Find a second joint PMF that is also compatible with and , and different from your answer in (a).

(c) What is the correlation type of and in (a) and (b)?

Random point in a square

A random point is chosen inside the square .

Write for the coordinates of the chosen point. Define the random variables by and .

(a) Find the marginal PDF of , namely .

(b) Find the marginal PDF of , namely .

(c) Find .

(d) Are and independent?

CLT and discrete wait time

Role a die repeatedly until you have rolled a one 3,000 times.

What (approximately) are the odds that you will have to roll more than 18,500 times before this happens?

Conditioning by a variable event

Roll a standard die, and let denote the number rolled.

Now toss a fair coin times. Let be the number of heads observed.

What is ?

Drawing cards with replacement

A standard deck of cards has 52 cards in 4 suits (hearts, spades, diamonds, clubs). A card is drawn and then replaced 8 times.

(a) The total number of hearts drawn is counted by the variable . Define the applicable distribution including parameters.

(b) We seek the probability that the heart drawn will be drawn in the first half. Define the applicable distribution, with parameters, and compute an expression for this probability.

Drawing marbles with replacement

An urn contains 3 blue marbles and 4 green marbles. A marble is drawn, and replaced, and another marble is drawn.

(a) Describe the sample space.

(b) What is the probability that the drawn marbles have different colors?

(c) Let be a random variable counting the number of blue marbles drawn. What is the name of the probability distribution of ? What is ?