Probability Problems for Scrabble, Rat Feeder, Coin Tossing Games, Blindfolded Musical Chairs, and R

A scrabble hand is a set of 7 tiles, each having one of the english uppercase letters on them, drawn uniformly at random from a bag of 100 tiles. The number of tiles of each letter are as follows:

E×12, A×9, I×9, O×8, N×6, R×6, T×6, L×4, S×4, U×4, D×4, G×3, B×2, C×2, M×2, P×2, F×2,H×2, V×2, W×2, Y×2, K×1, J×1, X×1, Q×1, Z×1

1. What is the probability that a scrabble hand contains the word HEXAGON ?

2. What is the probability that a scrabble hand contains the word GARBAGE ?

3. What is the probability that a scrabble hand contains the word APPLE ?

Hint: Review the notes on uniform probability spaces (Oct 10, Section 5.4)

A rat feeder is essentially a straw whose diameter is just large enough for 1 (medicine) pill or  1 (food) pellet, but is long enough to hold many pills and pellets. The pill and pellets are put in at one of the feeder and come out the other end (when the rat presses a pedal) in the same order they were put in.

Suppose we place 25 identical pellets and 4 identical pills uniformly at random into a rat feeder. The rat then comes and consumes one item from the feeder and then consumes another item from the feeder.

1. Let be the event " is a pellet" and let be the event " is a pill".

2. What is ?

3. What is ?

4. Are the events and independent? In other words, is ?

x1

x2

A x1 B x2

Pr(A ∩ B)

Pr(A ∪ B)

A B Pr(A ∩ B) = Pr(A) ⋅ Pr(B)

Consider the following coin tossing games. For each one, compute the probability that you win the game. For each question, the answer is a rational number so you should give this number exactly and give a decimal approximation of it as well.

1. You toss a fair coin twice and win if it comes up heads at least once.

2. You toss a fair coin 10 times and win if comes up heads at least five times.

3. You toss a fair coin twice and win if it comes up heads exactly once.

4. You toss a fair coin 10 times and win if comes up heads exactly five times.

We are playing a game of blindfolded musical chairs with 20 blindfolded people and 40 chairs. When the music stops each person picks a chair uniformly at random and sits on it.

1. What is the probability that some chair has at least two people sitting on it?

2. What is the probability that some chair has at least three people sitting on it?

We independently pick two random numbers and from the set . (Note: Independence means we may pick .)

1. What is the probability that and are both even?

2. Suppose I tell you that at least one of or is even. What is the (conditional) probability that and are both even?

3. What is the probability that at least one of or is equal to ?

4. Suppose I tell you that at least one of or is even. What is the probability that at least one of or is equal to ?

5. Suppose I tell you that at least one of or is even. What is the probability that at least one of or is equal to ?

R1 R2 {1,…, 1000}

R1 = R2

R1 R2

R1 R2

R1 R2

R1 R2 1000

R1 R2

R1 R2 1000

R1