I'm with Melody.

For the 20 ball first part, the number of three ball combinations is 20C3 = 1140.

The number of ways in which we miss with all three numbers is 17C3 = 680.

The number of ways in which we have 1 hit and three misses is 3*17C2 = 408.

The number of ways in which we have 2 hits and 1 miss is 3C2*17 = 51.

The number of ways in which we have three hits is 1.

(Check that all possibilities have been covered, 680 + 408 + 51 + 1 = 1140.)

That gets us the probabilities,

zero hits 680/1140 = 0.59649 (5 dp),

1 hit 408/ 1140 = 0.35789 (5 dp),

2 hits 51/1140 = 0.04474 (5 dp),

3 hits 1/1140 = 0.00088 (5 dp).

An alternative method is to calculate the probabilities directly.

Using H for a hit and M for a miss, the probability of three misses MMM is (17/20)*(16/19)*(15/18) = 0.59649.

The probability of the sequence HMM is (3/20)*(17/19)*(16/18) = 0.119298, and since both of the other 1 hit sequences MHM and MMH have the same probability, the probability of 1 hit will be 3*0.119298 = 0.35789 (5 dp).

The probabilty of the two hit sequence HHM is (3/20)*(2/19)*(17/18) = 0.014912, and since both of the other two hit sequences HMH and MHH have the same probability, the probability of two hits will be 3*0.014912 = 0.04474 (5 dp).

The probability of three hits will be (3/20)*(2/19)*(1/18) = 0.00088 (5 dp).

That means that the probability of at least two hits (two matches) is 0.04474 + 0.00088 = 0.04562.

The probability of not winning that way will be 0.59649 + 0.35789 = 0.95438.

The probability of winning with the superball is 0.1 and not winning with the superball is 0.9.

There are four possible outcomes,

(1) not getting at least two matches and not picking the superball : probability 0.95438* 0.9 = 0.85894,

(2) getting at least two matches, but not the superball : probability 0.04562*0.9 = 0.04106,

(3) not getting at least two matches but do pick the superball : probability 0.95438*0.1 = 0.09544,

(4) getting at least two matches and picking the superball : probability 0.04562*0.1 = 0.00456,

(the probabilities summing to 1, as they should).

That means that the probability of winning is 0.04106 + 0.09544 + 0.00456 (or 1 - 0.85894) = 0.14106.

Tiggsy.