In the SuperLottery, three balls are drawn (at random) from ten white balls numbered from to , and one SuperBall is drawn (at random)

You can win in any of four ways:

 

1) Getting two of the three winning white balls and the red SuperBall.

    Two of the three wining white balls and one out of seven losing white balls:  

           3C2·7C1 / 10C3  =  3·7/120  =  7/40

    Combine this with getting the red SuperBall  =  1/10     --->     7/400

 

2) Getting two of the three white balls and not getting the red SuperBall:

    The  7/40  from above combined with not getting the red SuperBall  =  9/10   --->   63/400

 

3) Getting all three of the white balls and the red SuperBall:

    Three of three winning white balls and none of the seven losing white balls:

          3C3·7C0 / 10C3  =  1·1/120  =  1/120

    Combine this with getting the red SuperBall  =  1/10     --->     1/200

 

4)  Getting all three of the white balls and not getting the SuperBall:

      The  1/120  from above combined with not getting the red SuperBall = 9/10   --->   3/400

 

 

Adding up all the probabilities, we get the answer of 3/16.

I think you mean that the question is:

 

In the SuperLottery, three balls are drawn (at random, without replacement) from ten white balls numbered from 1 to 10, and one SuperBall is drawn (at random) from ten red balls numbered from 11 to 20. When you buy a ticket, you choose three numbers from 1 to 10 and one number from 11 to 20.

If the numbers on your ticket match at least two of the white balls or match the red SuperBall, then you win a super prize. What is the probability that you win a super prize?

 

Answer:

 

We compute the complement: we'll count the number of losing tickets.

 

To have a losing ticket, you must have at most one correct white ball, and miss the SuperBall.

You miss all 3 white balls if your ticket contains 3 of the 7 white numbers that were not drawn, so there are \(\dbinom{7}{3}=\dfrac{7\cdot6\cdot5}{6}=35\) possibilities.

You hit 1 white ball and miss the others if your ticket contains 1 of the 3 white numbers that were drawn and 2 of the 7 white numbers that were not drawn, so there are \(3\dbinom{7}{3}=\dfrac{3\cdot7\cdot6}{2}=63\) possibilities.

You miss the SuperBall if you have one of the 9 red numbers that were not drawn.

Therefore, there are \((35+63)\cdot9=882\) losing tickets.

Hence, there are \(1200-882=318\) winning tickets, and your probability of winning a super prize is

 

\(\dfrac{318}{1200}=\boxed{\dfrac{53}{200}}.\)
 

 

Note: We can approach this problem using direct counting, but there are a number of cases:

1.  Matching 3 white balls with any super ball.
2. Matching 2 white balls with any super ball.
3. Matching 1 white ball and matching the super ball.
4. Matching 0 white balls and matching the super ball.

Given the numerous cases here, a complementary counting approach is a faster approach.