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# Tag Archives: JCM_math230_HW2_S13

## Finding a good phone

At the London station there are three pay phones which accept 20p coins. one never works, another works, while the third works with probability 1/2. On my way to London for the day, I wish to identify the reliable phone, so that I can use it on my return. The station is empty and I have just three 20p coins. I try one phone and it doesn’t work. I try another twice in succession and it works both times. What is the probability that this second phone is the reliable one ?

[Suhov and Kelbert, p.10, problem 1.9]

## 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

## Polya’s urn

An urn contains \(4\) white balls and \(6\) black balls. A ball is chosen at random, and its color is noted. The ball is then replaced, along with \(3\) more balls of the same color. Then another ball is drawn at random from the urn.

- Find the chance that the second ball drawn is white.
- Given the second ball drawn is white, what is the probability that the first ball drawn is black ?
- Suppose the original contents of the urn are \(w\) white and \(b\) black balls. Also after drawing a ball we replace with \(d\) balls of the same color. What is the probability that the second ball drawn is white (it should be \(\frac{w}{w+b}\) )?

[Pitman page 53. Problem 2]

## The chance of being English

English and American spellings are rigour and rigor, respectively. An English speaking guest staying at a Paris hotel writes the word and chose a letter at random from his spelling. The letter turns out to be a vowel. (that is any of : e,a,i,o,u). If 40% of the English speaking guests are American and 60% are English, what is the probability that the writer is American ?

[Ross, p. 107 #29]

## 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.

## Cards: Independence

A card is selected at random from a deck of 52 playing cards. If \(E\) is the event that the card is a King and \(F\) is the event that it is a heart. Show that \(E\) and \(F\) are independent events