Reference no: EM132328937

What's The Best Bet

Most areas of mathematics developed from uplifting and noble research. However there was notable exception. Probability theory, one of the most important areas of all, had its origins is vice.

The simplest bet of all is on the toss of a coin. I will give you $10 if it's heads, you give me $10 if it's tails. This is so obviously a ‘fair' bet that the maths is taken for granted. Nonetheless, let's go through the basic mathematics, which can be applied to more complicated bets later on.

Everyone knows that the chance of a head on the toss of a coin is "Fifty-fifty". This is the language of odds with which people are most comfortable. There are, however, many different ways of describing a probability, all of which mean exactly the same thing. The chance of tossing a head on an unbiased coin can be expressed as any of the following:

• Fifty-fifty
• 1 in 2
• ½ - mathematicians often quote probabilities as fraction.
• 0.5 - mathematicians also like using decimal.
• 50% - percentages are favoured by weather forecasters for some reason.

What they all mean is that if you toss an ordinary coin 100 times, you expect it to come up heads 50 times. Sometimes more, sometimes fewer, but on average 50.

To work out how much you expect to make from a bet, you need to look at how much you stand to win or lose for each possible outcome, and the chance of each outcome happening. In the example of heads and tails with $10 at stake, suppose that you call heads. Here are the possible outcomes:

Result	Chance of it happening (P)	How much you will win (W)	P × W
Heads	½	$10	$5
Tails	½	-$10	-$5

Is this bet worth taking? This is where the final column (P × W) comes in useful. If you add up the column, it gives you the expected value of the bet. In this case, the value is $0, which means that on average you should end up no worse off either, which makes this better than most bets available to you on the market! This is because in most cases, the value will end up with a negative number.

And as far as the gambling industry is concerned, that is how it should be. The whole point of gambling is to give you the prospect of making a fantastic return on your investment of a bet in a single turn, while ensuring that in the long run the organiser will make a profit.

(Source: Why Do Buses Come in Threes? by Rob Eastaway & Jeremy Wyndham © 1998 John Wiley & Sons, Inc.)

ANSWER ALL THE QUESTIONS

Base on the article that you have read, answer the following questions.

(1) Calculate the expected value for the following games:

(a) You roll a fair die, and the return as follow:

Result	How much you win
Throw 6	$25
Don't throw 6	- $6

(b) You bet $1 (this money will not be returned) on each of the three horses in 3 different races, and the return as follow:

Result	Chance of it happening	How much you win
White horse wins	½	$2
Black horse wins	1/3	$3
Brown horse wins	¼	$4

(c) You draw 2 cards from a deck of 52 cards and the return as follow:

Result	How much you win
Blackjack or Double Aces	$50
Total is more than 16 (excluding Blackjack or Double Aces)	$10
None of the above	- $5

(2) A box contains 99 discs with label 1 - 99 (each disc with different number). 4 men and 4 women, each of them randomly draw a disc from the box. All of them are ranked according to their scores on the disc that they drawn. Let X be the highest ranking achieved by a man, where X = 1, 2, 3, ...., 8.

(a) Find the variance of X

(b) Find E(X), given that X greater than 2.

(3) From the long experience, a staff in a casino found that the duration a visitor spends in the casino per visit is normally distributed with mean 10k minutes and standard deviation k minutes.

(a) If two visitors are selected randomly, find the probability that

(i) the first visitor spends 20% more time than the second visitor in the casino.

(ii) the total time of the two visitors in the casino is more than 18k minutes.

(b) Peter and Janet want to meet at the casino about 8:00 pm. Given that Peter arrives at a time uniformly distributed between 7:15 pm to 8:15 pm and Janet independently arrives at a time uniformly distributed between 7:40 pm and 8:10 pm. Find the probability that

(i) both of them arrive before 8:00 pm.

(ii) only one of them arrives before 8:00pm given that Janet arrives after 7:45 pm.

(iii) What is the probability that the first to arrive waits no longer than 5 minutes?

(c) The time between arrivals of customers at the cashier counter of the casino is an exponential random variable with a mean of 1 minute. What is the probability that

(i) more than five customers arrive in 2 minutes?

(ii) the time until the 4th customer arrives is less than 3 minutes?

(4) Alex and Ben play a game. In the game, both of them roll three fair dice. Let X₁, X₂ and X₃ be the score of the three dice that they roll and A and B be the score that Alex and Ben obtain respectively. Given that

A = min (X₁, X₂, X₃) and B = max (X₁, X₂, X₃)

(a) Find the CDF for B.

(b) Compute the moment generating function for A.

(c) Find

(i) P[( A ≥ 2) ∩ (B ≤ 2)]

(ii) P[( A ≥ 2) ∪ (B ≤ 2)]

(iii) P(| A - B |< 2)

(d) In this game, the game will end when A = 6 or B = 1 whichever comes first. Let N represent the number of rounds the game has played before ending, find E(N).

(5) In a lottery that contains 5 digits, the players will win a large prize when they pick 5 digits that match, in the correct order that selected by a random mechanical process. A smaller prize is won if only 4 digits are matched.

(a) How many 5-digit numbers that can win the smaller prize?

(b) What is the probability that a player wins the large prize?

(c) What is the probability that a player not wins any prize?

(d) What is the probability that a player not wins any prize given that the first digit is matched with the number that selected randomly?