At the school playground, students can check out 20 red playground balls to play with.Eight of the balls are of superior quality and bounce well enough to support a decent game of double four square.If we need two of the superior-quality balls to play the game and the four players each randomly select a playground ball to check out, what is the probability that we will be able to have a decent game (i.e., have at least two superior-quality balls)?
A) 0.4654
B) 0.3814
C) 0.5346
D) 0.5248

Question

Quizplus · Accepted Answer

The total number of ways the four players can select the playground balls is 20 choose 4, or (20!)/(4!16!), which is equal to 4,845. 
If we want to have at least two superior-quality balls, we can first calculate the number of ways we can select exactly two superior-quality balls and two non-superior quality balls. There are 8 choose 2 ways to select the two superior-quality balls and 12 choose 2 ways to select the two non-superior quality balls. This is equal to (8!)/(2!6!) * (12!)/(2!10!), or 28 * 66, which is equal to 1,848. 
We can also select three or four superior-quality balls. There are 8 choose 3 ways to select three superior-quality balls and 12 choose 1 ways to select one non-superior quality ball. This is equal to (8!)/(3!5!) * (12!)/(1!11!), or 56 * 12, which is equal to 672. There is only 1 way to select all four superior-quality balls. 
Thus, the total number of ways we can have at least two superior-quality balls is 1,848 + 672 + 1 = 2,521. 
Therefore, the probability of having at least two superior-quality balls is 2,521/4,845, which is approximately 0.5346.

At the School Playground, Students Can Check Out 20 Red