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Finding x distribution table and mathematical expectation
1)

When x=0, p = (1-1/2) 3 =1/8.

When x= 1, p = 3 * (1-1/2) 2 *1/2 = 3/8.

When x=2, p = 3 * (1-1/2) * (1/2) 2 = 3/8.

When x=3, p = (1/2) 3 = 1/8.

So the probability distribution list is

x 0 1 2 3

P 1/8 3/8 3/8

2)

B Probability of hitting the target three times p' = (2/3) 3 = 8/27.

So the probability of hitting twice at most is P= 1-8/27= 19/27.

3)

In two cases, A hit twice and B missed; Or a hits three times and b hits once.

p 1=3/8*( 1/3)^3= 1/24

p2= 1/8*3*(2/3)*( 1-2/3)^2= 1/36

So the total probability is P 1+P2=5/72.

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