Most areas of mathematics developed from uplifting and noble research However there notable exception. Probability theory, one of the most important areas of all, had its origins is was 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 1later 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 tossing a head on an unbiased coin can be expressed as any of the following probability, all of which mean exactly the same thing The chance of a Fifty-fifty 1 in 2 -mathematicians often quote probabilities 0.5 mathematicians also like using decimal 50%-percentages as fraction. 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: Chance of it happening How much vou will win Result P x W (P) (W) $10 -$10 Heads $5 $5 Tails Is this bet worth taking? This is where the final column (P x 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 negative number a And as far as the gambling industry is concemed, that is how it should be. The whole point of gambling is to give you the prospect of making a fantastic retun on your investment of a bet in a single tum, while ensuring that in the long run the organiser will make a profit the article that you have read, answer the following questions Base on Calculate the expected value for the following games (a) (1) You roll a fair die, and the retum as follow How much you win $25 - $6 Result Throw 6 Don't throw 6 [2 marks] You bet $1 (this money will not be returned) different races, and the return as follow (b) on each of the three horses in 3 Chance of it happening How much you win $2 Result White horse wins Black horse wins $3 $4 1/3 Brown horse wins 1/4 [3 marks] You draw 2 cards from a deck of 52 cards and the return as follow (c) How much vou win Result Blackjack or Double Aces Total is more than 16 $50 $10 (excluding Blackjack or Double Aces) $5 None of the above (Assume the face value of an ace is 11) [4 marks] Most areas of mathematics developed from uplifting and noble research However there notable exception. Probability theory, one of the most important areas of all, had its origins is was 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 1later 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 tossing a head on an unbiased coin can be expressed as any of the following probability, all of which mean exactly the same thing The chance of a Fifty-fifty 1 in 2 -mathematicians often quote probabilities 0.5 mathematicians also like using decimal 50%-percentages as fraction. 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: Chance of it happening How much vou will win Result P x W (P) (W) $10 -$10 Heads $5 $5 Tails Is this bet worth taking? This is where the final column (P x 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 negative number a And as far as the gambling industry is concemed, that is how it should be. The whole point of gambling is to give you the prospect of making a fantastic retun on your investment of a bet in a single tum, while ensuring that in the long run the organiser will make a profit the article that you have read, answer the following questions Base on Calculate the expected value for the following games (a) (1) You roll a fair die, and the retum as follow How much you win $25 - $6 Result Throw 6 Don't throw 6 [2 marks] You bet $1 (this money will not be returned) different races, and the return as follow (b) on each of the three horses in 3 Chance of it happening How much you win $2 Result White horse wins Black horse wins $3 $4 1/3 Brown horse wins 1/4 [3 marks] You draw 2 cards from a deck of 52 cards and the return as follow (c) How much vou win Result Blackjack or Double Aces Total is more than 16 $50 $10 (excluding Blackjack or Double Aces) $5 None of the above (Assume the face value of an ace is 11) [4 marks]
