Probability problems and solutions

Monte Hall Problem

Consider the following version of the Monte Hall Problem:
There are four doors numbered 1,…,4. Three of these doors have goats behind them. One door has a sports car. You want to win the sports car.
You pick one door, i uniformly at random and put a chalk mark on it. That door stays closed for now.
Monte Hall opens a door j, with j≠i, showing you a goat.

You get to pick any of the three unopened doors in {1,2,3,4}?{j} and keep whatever is behind it.
Answer the following questions
1. Suppose you decide to stick with your first choice, i. What is the probability that you win the sports car?
2. Suppose you decide to choose one of three unopened doors {1,2,3,4}?{j} uniformly at random. What is the probability that you win the sports car?
3. Suppose you decide to choose one of the two unopened and unmarked doors {1,2,3,4}?{i,j} uniformly at random. What is the probability that you win the sports car?
4. Estimating Genetic Diseases
One out of 25 healthy people carries a single gene for cystic fibrosis (CF), these people are called carriers and healthy people without a CF gene are called non-carriers. A uniformly-chosen random healthy person has probability 1/25 of being a carrier.
A person with two CF genes is not healthy; they are sick (with cystic fibrosis). The child of a carrier has probability 1/2 of inheriting a CF gene from that parent. The child of two carriers inherits each of their parents CF genes independently, so the child has probability 1/4 of having CF.
1. Two uniformly-chosen random healthy people have a child together. What is the probability that the child has CF?
2. Two uniformly-chosen random healthy people have a child together. What is the probability that the child is a healthy non-carrier?
3. A carrier has a child with a uniformly-chosen random healthy person. What is the probability that the child has CF?
4. A carrier has a child with a uniformly-chosen random healthy person. What is the probability that the child is a (healthy) carrier?
5. Two uniformly-chosen healthy people have a baby. A quick blood test, administered minutes after birth, shows that the baby has at least one CF gene, but gives no other information. What is the probability that the baby has CF?
(This question is based a personal story during which I learned that there are genetic counsellors, a big part of whose job is computing these conditional probabilities to help guide prospective parents in making informed decisions.)
5. Sampling With and Without ReplacementWe have a cooler containing 2 cider bottles and n−2 beer bottles. We can sample from this cooler in two different ways:
with replacement: We take a uniformly-random bottle from the cooler, look at it and put it back.
without replacement: We take a uniformly-random bottle from the cooler, drink it, and throw the empty bottle into the recyling bin.Suppose we repeatedly sample from the cooler and let X be the number of samples up to and including the first cider bottle.
1. What is E(X) if we use sampling with replacement?
2. What is E(X) if we use sampling without replacement?(This question highlights a fact that beer, whisky, and wine connoisseurs know intuitively: sampling without replacement is more fun, but makes things harder to compute!)
6. Finding a Healthy Subring
A group of n people p0,…,pn−1 are standing around in a circle holding hands. When we talk about these people, we take indices modulo n so that, for example pn+5=p5. s of these people are sick, the rest are healthy.
1. Let k2. Argue that if we have n=70 people standing around in a circle and s=39 of them are sick, then there always exists 7 consecutive people pi,…,pi+6 at least 4 of whom are healthy. (This should be a bit surprising, since 4/7>1/2 of these people are healthy but 39/70>1/2 of all people are sick.