The Palace of Peace and Accord in Astana, Kazakhstan, was built in 2006. The building has a square right pyramid structure. The
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Let be the random variable representing the winnings you get for playing the game. Then
First thing to do is determine the probability of each of the above events. You roll two dice, which offers 6 * 6 = 36 possible outcomes. You find the probability of the above events by dividing the number of ways those events can occur by 36.
- The sum is odd if one die is even and the other is odd. This can happen 2 * 3 * 3 = 18 ways. (3 ways to roll even with the first die, 3 ways to roll odd for the die, then multiply by 2 to count odd/even rolls)
- The sum is 4 if you roll (1, 3), (2, 2), or (3, 1), and the sum is 8 if you roll (2, 6), (3, 5), (4, 4), (5, 3), or (6, 2). 8 ways.
- The sum is 2 if you roll (1, 1), and the sum is 12 if you roll (6, 6). 2 ways.
- There are 36 total possible rolls, from which you subtract the 18 that yield a sum that is odd and the other 10 listed above, leaving 8 ways to win nothing.
So the probability mass function for this game is
The expected value of playing the game is then
or about $7.89.
Fraction with variable x in numerator and 6 in the denominator can written in the algebraic expression as
Let's assume some number is n.
So, the sum of six and some number can be written as 6+n.
The product of six and some number can be written as 6*n.
The quotient of some number and six will be n/6.
The difference of six and some number will be 6-n.