Calculating conditional probability from joint probability distribution table

Question

I am working on the following homework problem, given the table below:

| a | b | c | p(a,b,c) |
|:-:|:-:|:-:|:--------:|
| 0 | 0 | 0 |   0.192  |
| 0 | 0 | 1 |   0.144  |
| 0 | 1 | 0 |   0.048  |
| 0 | 1 | 1 |   0.216  |
| 1 | 0 | 0 |   0.192  |
| 1 | 0 | 1 |   0.064  |
| 1 | 1 | 0 |   0.048  |
| 1 | 1 | 1 |   0.096  |

a) Determine whether a and b are independent: p(a ∩ b) = p(a)p(b)

b) Determine whether a and b are conditionally independent on c: p(a ∩ b | c) = p(a | c)p(b | c)

My thought process was below:

1. First, the probability of a and b

p(a) = all values where a = 1 p(a) = 0.192 + 0.064 + 0.048 + 0.096 = 0.4

p(b) = all values where b = 1 p(b) = 0.048 + 0.216 + 0.048 + 0.096 = 0.408

2. First, the probability of p(a ∩ b)

p(a ∩ b) = all values where a = 1 and b = 1 p(a ∩ b) = 0.048 + 0.096 = 0.144

3. Compare values 0.144 = ((0.4)(0.408)) = NOT TRUE

Therefore, a and b are dependent events

Now, moving onto b):

1. Calculate all relevant values:

p(a) = all values where a = 1 p(a) = 0.192 + 0.064 + 0.048 + 0.096 = 0.4

p(b) = all values where b = 1 p(b) = 0.048 + 0.216 + 0.048 + 0.096 = 0.408

p(c) = all values where c = 1 p(c) = 0.144 + 0.216 + 0.064 + 0.096 = 0.52

p(a,c) = all values where a = 1 and c = 1 p(a,c) = 0.064 + 0.096 = 0.16 p(a|c) = 0.16 / 0.52 = 0.31

p(b, c) = all values where b = 1 and c = 1 p(b, c) = 0.216 + 0.096 = 0.312 p(b|c) = 0.312 / 0.52 = 0.6

Thus p(a|c)p(b|c) = (0.31)(0.6) = 0.119

Knowing p(a,b,c) = 0.192, then 0.192 / 0.52 = 0.369

Therefore we know that a and b are not conditionally independent given the values of c

Therefore, my questions are:

1. Is this work correct? If not, what is wrong?
2. Why am I using a/b/c = 1 instead of 0?

Answer 1

Let us convert first, the given information in the table to given probabilities

P(a) = 0.192+0.064+0.048+0.096 = 0.4

P(b)= 0.048+0.216+0.048+0.096 = 0.408

P(c) = 0.144+0.216+0.064+0.096 = 0.52

Consider p(a ∩ b) = p(a)p(b) = 0.4*0.408= 0.1632 ...... (1)

But P(a ∩ b) = 0.048+0.096= 0.144 ....... (2) (From the table)

Since (1) and (2) are not equal, we conclude that a and b are not

independent.

Consider now p(a ∩ b/c) = p(a∩b∩c)/P(c)= 0.096/0.52 = 0.185...(3)

Consider P(a/c) = p(a∩c)/P(c)=(0.064+0.096)/0.52 = 0.16/0.52 = 0.308

Consider P(b/c) = p(b∩c)/P(c)=(0.216+0.096)/0.52 = 0.312/0.52 = 0.0.6

So, P(a/c)*P(b/c) = 0.308*0.6 = 0.1848 = 0.185 .... (4)

Since (3) and (4) are equal, we can say that

p(a ∩ b | c) = p(a | c)p(b | c)

So, a and b are conditionally independent on c