Probability: What is it and how do you calculate it?

Considering the image given, it has four different colored binds with a particular number of cubes with random numbers. Several rules can be set from the setting of these cubes and their respective bins for a particular game of chance of any interest. Assuming that a contestant picks only a single cube from the bins, they are not told which bin contains the required cube, and that there is no double selection, four rules can be created for each as follows:

Game 1: A contestant will win when they pick #7 cube which is green from the green bin

Game 2: A contestant will win when they pick #3 cube which is yellow from the blue bin.

Game 3: A contestant will win when they pick #6 which is blue from the red bin

Game 4: A contestant will win when they pick #15 which is yellow from the yellow bin

To calculate probabilities of two of these games, the main consideration is the total number of the required cube and the total cubes in the bins (Myerson, 2013). Considering Game 1 and Game 4 the respective probabilities can be calculated;

Game 1 = required cube is = 7 which is green

= total number of cubes = 24

= total number of green #7 = 1

Probability = picking a green #7 = 1/24 = 0.04167

Game 4 = requires #15 which is yellow

= total number of yellow #15 = 1

= total number of cubes = 24

Probability = 1/24 = 0.04167

Supposing that the cost to play per game is $1.00 for the first game and $2.00 for the second game, and that there are 40 and 60 players for these games respectively, the net expected value can be calculated. However, this would be possible when the net earnings are known as in the case. For Game 1, the net earnings are $1.00 per win and assuming that the 40 players win, the House expected value would be $40 (40 * $1 = $40). On the other hand, for 60 players, the House expected value will be $120 (60 * $1.5 = $120). Considering these small probabilities of winning, these payouts seem appropriate. However, adopting package games would encourage more people to play the game. Buying a package of games rather than one game at a time increases the probability of selecting the required cube (Myerson, 2013). Consequently, this would increase then House expected value.

References 

Myerson, R. B. (2013). Game theory . Harvard university press.