One of the wagers in roulette is to bet that the ball will stop on a number that is a multiple of 3. (Both 0 and 00 are not included.) If the ball stops on such a number, the player wins double the amount bet. If a player bets $1, compute the player's expectation.
There are a total of 36 numbers in roulette (excluding 0 and 00), out of which 12 are multiples of 3.
The probability of the ball stopping on a number that is a multiple of 3 is therefore 12/36 or 1/3.
If the player wins, they receive double the amount bet, which is $1 in this case.
So, the player has a 1/3 probability of winning $2 and a 2/3 probability of losing $1.
To compute the player's expectation, we multiply the probability of each outcome by the corresponding payout and sum them up.
Expectation = (Probability of winning * Amount won) + (Probability of losing * Amount lost)
= (1/3 * $2) + (2/3 * -$1)
= $2/3 - $2/3
= $0
The player's expectation is $0, which means they can expect to break even in the long run.
What is a rational number with a denominator of 7 that is between √7 and √8. Write your answer as an improper fraction.
To find a rational number between √7 and √8 with a denominator of 7, we can consider the square root of 7 as approximately 2.64575 and the square root of 8 as approximately 2.82843.
Now, we need to find a rational number x such that √7 < x < √8, and the denominator of x is 7.
Multiplying both sides of the inequality by 7, we have:
√7 * 7 < x *7 < √8 * 7
7√7 < 7x < 7√8
To obtain an integer value for the denominator, we will choose the smallest integer that x * 7 is greater than or equal to 7√7.
Let's try x = 2. Let's check if 2 * 7 = 14 is greater than or equal to 7√7.
If we square 14 - 7√7, it should be greater than or equal to 0.
(14 - 7√7)^2 = 14^2 - 2 * 14 * 7√7 + (7√7)^2
= 196 - 98√7 + 49 * 7
= 196 - 98√7 + 343
= 539 - 98√7
Since 539 - 98√7 is positive, x = 2 is not a suitable choice.
Let's try x = 3.
(21 - 7√7)^2 = 21^2 - 2 * 21 * 7√7 + (7√7)^2
= 441 - 294√7 + 49 * 7
= 749 - 294√7
Since 749 - 294√7 is positive, our choice of x = 3 is also not suitable.
Let's try x = 4.
(28 - 7√7)^2 = 28^2 - 2 * 28 * 7√7 + (7√7)^2
= 784 - 392√7 + 49 * 7
= 1232 - 392√7
Since 1232 - 392√7 is positive, x = 4 is also not a suitable choice.
Let's try x = 5.
(35 - 7√7)^2 = 35^2 - 2 * 35 * 7√7 + (7√7)^2
= 1225 - 490√7 + 49 * 7
= 2107 - 490√7
Since 2107 - 490√7 is positive, our choice of x = 5 is also not suitable.
Finally, let's try x = 6.
(42 - 7√7)^2 = 42^2 - 2 * 42 * 7√7 + (7√7)^2
= 1764 - 588√7 + 49 * 7
= 2744 - 588√7
The expression 2744 - 588√7 is greater than 0, which means x = 6 is a suitable choice.
Therefore, a rational number with a denominator of 7 that is between √7 and √8 is 42/7 = 6.
So, the improper fraction is 42/7.
Describe the relationship between rational numbers and whole numbers. Are all rational numbers also whole numbers? Are all whole numbers also rational numbers? Give examples to support your answer.
Thank you.
Rational numbers and whole numbers are closely related, but they are not the same.
Rational numbers include all numbers that can be expressed as a fraction, where the numerator and denominator are both integers. Whole numbers, on the other hand, are positive integers (0, 1, 2, 3, ...) and their negatives (-1, -2, -3, ...).
Not all rational numbers are whole numbers. Whole numbers are a subset of rational numbers, but rational numbers can also include fractions and decimals. For example:
- The fraction 3/1 is a rational number, but it is also a whole number because the numerator and denominator are both integers.
- The fraction 4/2 is another rational number, but it simplifies to 2, which is a whole number.
However, there are many rational numbers that are not whole numbers. For example:
- The fraction 5/2 is a rational number, but it is not a whole number because the numerator and denominator are not both integers.
- The decimal 0.333... (or 1/3) is a rational number, but it is not a whole number.
All whole numbers are rational numbers because they can be expressed as fractions where the denominator is 1. For example, the whole number 5 can be written as the fraction 5/1, which is rational.
In summary, while all whole numbers are rational numbers, not all rational numbers are whole numbers. Rational numbers encompass a broader range of numbers, including whole numbers, fractions, and decimals.