A gambler bets on one of the integers from 1 to 6. Three fair dice are then rolled. If the gambler’s number appears times (), he wins $ . If his number fails to appear, he loses $1. Calculate the gambler’s expected winnings

### Question of the Day #18

Here is a very very interesting question involving probability that a student saw in her tutorial and asked me. Here it is 🙂

A student is concerned about her car and does not like dents. When she drives to school, she has a choice of parking it on the street in one space, parking it on the street and taking up two spaces, or parking in the lot.

If she parks on the street in one space, her car gets dented with probability 0.1.

If she parks on the street and takes two spaces, the probability of a dent is 0.02 and the probability of a $15 ticket is 0.3.

Parking in a lot costs $5, but the car will not get dented.

If her car gets dented, she can have it repaired, in which case it is out of commission for 1 day and costs her $50 in fees and cab fares. She can also drive her car dented, but she feels that the resulting loss of value and pride is equivalent to a cost of $9 per school day.

She wishes to determine the optimal policy for where to park and whether to repair the car when dented in order to minimize her (long-run) expected average cost per school day. What should the student to maximise her utility (minimise her cost)?

This is an interesting question, I guess its good to know some JCs are trying to introduce decision making process in teaching probability.

I’ll post a solution here soon. But to start off, we observe that we have two states here and student has 4 decisions. Have fun! 🙂

### Combinatorics related articles

Here is a compilation of all the Combinatorics articles KS has done. Students should read them when they are free to improve their mathematics skills. They will come in handy! 🙂

### 2002 A-level H2 Mathematics Paper 2 Question 30 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(a)

(i)

Alternatively,

(ii)

When , required probability

(b)

Key GC with either sequence or use table functions.

We want to solve

Plot and check for the x that gives a corresponding value that is less than

(c)

When , probability that all 21 people have different birthdays

Probability that all 21 people have different birthdays

Thus, when , probability that at least two of the people are the same birthday (more than )

### KS Comments:

A very interesting probability question here. This show show counter intuitive probability is! The results shows that the we can expect to find someone with the same birthday in a room of 23, more than half the time.

### 2010 A-level H2 Mathematics (9740) Paper 2 Question 7 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)

(ii)

(iii)

P(B’|A)

(iv)

since A and C are independent

(v)

### 2015 A-level H2 Mathematics (9740) Paper 2 Question 9 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)

since A and B are independent.

(ii)

(iii)

If ,

(maximum)

If ,

(minimum)

### Financial Engineering (I)

Before attempting to read what we have here, students should revise their basic probability and linear algebra first.

Financial Engineering (I) #1 – Overview

Financial Engineering (I) #2 – Introduction to No Arbitrage

Financial Engineering (I) #3 – Interest rates and fixed income instruments

Financial Engineering (I) #4 – Floating Rate Bonds and Term Structure of Interest Rates

Financial Engineering (I) #5 – Forward Contracts

Financial Engineering (I) #6 – Swaps

Financial Engineering (I) #7 – Futures

Financial Engineering (I) #8 – Options

Financial Engineering (I) #9 – Options Pricing

Financial Engineering (I) #10 – The 1-Period Binomial Model

Financial Engineering (I) #11 – Option Pricing in the 1-Period Binomial Model

Financial Engineering (I) #12 – The Multi-Period Binomial Model

Financial Engineering (I) #13 – Pricing American Options

Financial Engineering (I) #14 – Replicating Strategies

Financial Engineering (I) #15 – Dividends, Pricing in the Binomial Model

Financial Engineering (I) #16 – Black-Scholes Model

Financial Engineering (I) #17 – Introduction to Term Structure Lattice Models

Financial Engineering (I) #18 – Cash Account and Pricing Zero-Coupon Bonds

Financial Engineering (I) #19 – Fixed Income Derivatives (1)

Financial Engineering (I) #20 – Fixed Income Derivatives (2)

Financial Engineering (I) #21 – The Forward Equation

Financial Engineering (I) #22 – Model Calibration

Financial Engineering (I) #23 – Pricing in a Black-Derman Toy Model

Financial Engineering (I) #24 – Modelling and Pricing Default-able bonds

Financial Engineering (I) #25 – Credit Default Swaps and Pricing Credit Default Swaps

Financial Engineering (I) #26 – Mortgage Mathematics and Mortgage-Backed Securities

Financial Engineering (I) #27 – Prepayment Risks and Pass-Throughs

Financial Engineering (I) #28 – Principal-Only and Interest Only Mortgaged-Backed Securities

Financial Engineering (I) #29 – Collateralised Mortgage Obligations

Financial Engineering (I) #30 – Pricing Mortgage-Backed Securities

### Geometric Brownian Motion

This is really important for anyone interested in Finance Modelling. As what the movie Wolf on Wall Street says:

They are referring to a geometric brownian motion.

Firstly, we will begin with the definitions.

We say that a random process, , is a geometric Brownian motion (*GBM*) if for all

where is a Standard Brownian Motion

Here is the drift and is the volatility. We write

Also note that

; This is a common technique for solving expectations.

. This is very useful for simulating security prices.

Consider

; Notice this expansion is similar to before.

This result tells us that the expected growth rate of is .

From the definitions of Brownian Motion introduced earlier, we extend them to Geometric Brownian motion.

1. Fix . Then are mutually independent.

2. Paths of are continuous as function of , meaning they do not jump.

3. For ,

So now lets try to do some modelling of stock prices as a geometric brownian motion.

Suppose . Clearly

1. for any

This tells us that the limited liability of stock price is not violated.

2. The distribution of only depends on s and not on $latex X_t.

We will look at the Black-Scholes option formula next time and will come back to review the geometric brownian motion for the underlying model.

### Introduction to Brownian Motion

Lets look at brownian motion now. And yes, its the same as what our high school teachers taught about the particles moving in random motion. Here, we attempt to give it a proper structure and definition to work with.

A Brownian Motion is a random process with parameters if

For , are mutually independent. This is often called the independent increments property.

For

is a continuous function of t.

We say that is a Brownian motion with drift and volatility .

For the special case of and , we have a standard Brownian motion. We can denote it with and assume that

If and then where is a standard brownian motion. Thus,

The next concept is important in finance, that is, Information Filtrations.

For any random process, we will use to denote the information available at time t.

– the set is then the information filtration.

– denotes an expectation conditional on time t information available.

Note: The independent increment property of Brownian Motion implies that any function of is independent of and that .

So let us do a bit of math to obtain for instance.

Using condition expectation identity, we have

### Review of Basic Probability

So I understand that I lost many readers for the Sampling uploads.It is a bit difficult to the intensive use of notations and also the need for statistics knowledge. So here I’ll review a bit of basic probability. The contents here will be basic, and will involve some H2 Mathematics Statistics too.

Conditional Expectations and Variances

Multivariate Normal Distribution

*The following three are in relation to Finance.*