Question

|    | a1   | a2   |
|----|------|------|
| b1 | 0.37 | 0.16 |
| b2 | 0.23 | ?    |

1. What is ?(?=?2,?=?2)P(A=a2,B=b2)?

2. Observing events from this probability distribution, what is the probability of seeing (a1, b1) then (a2, b2)?

3. Calculate the marginal probability distribution, ?(?)P(A).

4. Calculate the marginal probability distribution, ?(?)P(B).

Answer 1

Solution:

Given:

		A
		a1	a2
B	b1	0.37	0.16
	b2	0.23	?

Part 1) What is P(A=a2,B=b2)?

To find this probability we need to use following property of probability.

Sum of all probabilities is 1.

Thus

P(A=a1,B=b1)+P(A=a1,B=b2)+P(A=a2,B=b1)+P(A=a2,B=b2)= 1

0.37+0.23+0.16+P(A=a2,B=b2)= 1

0.76+P(A=a2,B=b2)= 1

P(A=a2,B=b2)= 1-0.76

P(A=a2,B=b2)= 0.24

Part 2)  Observing events from this probability distribution, what is the probability of seeing (a1, b1) then (a2, b2)?

That is:

P(A=a1,B=b1) =.......?

and

P(A=a2,B=b2)= .....?

		A
		a1	a2
B	b1	P(A=a1,B=b1) = 0.37	P(A=a2,B=b1)0.16
	b2	P(A=a1,B=b2) = 0.23	P(A=a2,B=b2)=0.24

Thus from above table:

P(A=a₁,B=b₁) = 0.37

P(A=a₂,B=b₂)= 0.24

Part  3. Calculate the marginal probability distribution, P(A).

To find marginal probability distribution of A, we add column probabilities. P(A=a1) = 0.37+0.23=0.6 and

P(A=a2) = 0.16+0.24=0.4

		A
		a1	a2
B	b1	0.37	0.16
	b2	0.23	0.24
Marginal probability distribution of A: P(A)		P(A=a1) = 0.6	P(A=a2) = 0.4	Total=1.0

Part 4) Calculate the marginal probability distribution, P(B)

To find marginal probability distribution of B, we add row probabilities:

P(B=b1) =0.37+0.16 = 0.53 and P(B=b2) = 0.23+0.24 = 0.47

Thus

		A		Marginal probability distribution of B: P(B)
		a1	a2
B	b1	0.37	0.16	0.53
	b2	0.23	0.24	0.47
				Total = 1