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Independence of two hearts ?
Consider a deck of 52 cards. Let \(A\) be the event that the first card is a heart. Let \(B\) be the event that the 51st card is a heart.
What is \(\mathbf{P}(A)\) ? What is \(\mathbf{P}(B)\) ? Are \(A\) and \(B\) independent ?
Ordered Random Variables
Suppose \(X\) and \(Y\) are two random variable such that \(X \geq Y\).
- For a fixed number \(T\), which would be greater, \(\mathbf{P}(X \leq T) \) or \(\mathbf{P}(Y \leq T) \).
- What if \(T\) is a random variable ? (If it helps you think about the problem, assume \(T\) takes values in \(\{1,\cdots,n\}\). )
Balls in a Box: Counting
A box contains 20 red balls and 30 black balls. Four balls are chosen without replacement. What is the chance that:
- all balls are red
- exactly three balls are red
- the first red ball appears on the last draw.
- the fist two balls are the same color
Poker Hands: counting
Assume that each of Poker hands are equally likely. The total number of hands is
\[\begin{pmatrix} 52 \\5\end{pmatrix}\]
Find the probability of being dealt each of the following:
- a straight flush ( all cards of the same suit and in order)
- a regular straight (but not a flush)
- two of a kind
- four of a kind
- two pairs (but not four of a kind)
- a full house (a pair and three of a kind)
In all cases, we mean exactly the hand stated. For example, four of a kind does not count as 2 pairs and a full house does not count as a pair or three of a kind.
Chance of an Accident.
An insurance company has 50% urban and 50% rural customers. If every year each urban customer has an accident with probability \(\mu\) and each rural customer has an accident with probability \(\lambda\). Assume that the chance of an accident is independent from year to year and from customer to costumer. This is another way to say, conditioned on being and urban or rural the chance of having an accident each year is independent.
A costumer is randomly chosen. Let \(A_n\) be the chance this customer has an accident in year \(n\). Let \(U\) denote the event that this costumer is urban and \(R\) the event that the customer is rural.
- Find \( \mathbf{P}(A_2|A_1) \).
- Are \(A_1\) and \(A_2\) independent in general ? Are there any conditions when it is true if not in general ?
- Show that \(\mathbf{P}(A_2|A_1) \geq \mathbf{P}(A_2) \).
To answer this question it is useful to know that for any positive \(a\) and \(b\), one has \( (a+b)^2 < 2(a^2 +b^2)\) as long as \(a \neq b\). In the case \(a = b\), one has of course \( (a+b)^2 = 2(a^2 +b^2)\). To prove this inequality, first show that \( (a+b)^2 +(a-b)^2= 2(a^2 +b^2)\) and then use that fact that \( (a-b)^2 >0 \). - Find the probability that a driver has an accident in the 3nd year given that they had one in the 1st and 2nd year.
- Find the probability that a driver has an accident in the \(n\)-th year given that they had one in all of the previous years. What is the limit as \(n \rightarrow \infty\) ?
- Find the probability that a diver is a urban diver given that they had an accident in two successive years.
Monty Hall
Consider the Monty Hall problem:
Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1 [but the door is not opened], and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?
(You have observed that week after week on this show, the host always opens a door to revival a goat before giving you the option to switch)
- Without using conditional probabilities work out the probability of winning if you switch versus if you don’t.
- Compute the same probabilities using conditional probabilities.
- Now assume that the host is cruel. He only offers you a switch when you have chosen the car. (You figure this out by watching the show on TV before going.) Now should you switch when offered the chance ?
- Now lets assume that every morning the host gets up and with probability \(p\) acts like the original host and with probability \(1-p\) he acts like the cruel host from the previous question.
- Now what is the chance of wining if you switch ?
- Now what is the chance of wining if you don’t switch ?
- For which \(p\) should you switch and which \(p\) should you not switch ?
- Now let us assume that host just picks a door randomly after you have picked yours. If there is a car behind it the came is over. If there is a goat, he offers you the chance to switch. Should you switch when you have the chance ? why ?
Leukemia Test
A new drug for leukemia works 25% of the time in patients 55 and older, and 50% of the time
in patients younger than 55. A test group has 17 patients 55 and older and 12 patients younger than 55.
- A uniformly random patient is chosen from the test group, and the drug is administered and it is a success. What is the probability the patient was 55 and older?
- A subgroup of 4 patients are chosen and the drug is administered to each. What is the probability that the drug works in all of them?
