Question

Suppose S = {p, q, r, s, t, u} and A = {p, q, s, t} and B = {r, s, t, u} are events.

x p q r s t u

p(x) 0.15 0.25 0.2 0.15 0.1

(a) Determine what must be p(s).
(b) Find p(A), p(B) and p(A∩B).
(c) Determine whether A and B are independent. Explain.
(d) Arer A and B mutually exclusive? Explain.
(e) Does this table represent a probability istribution of any random variable? Explain.

Answer 1

Question (a)

x	p	q	r	s	t	u
P(x)	0.15	0.25	0.2	P(s)	0.15	0.1

Sum of all the probabiliites should be 1 for a radom variable

So P(x) = 1

0.15 + 0.25 + 0.2 + P(s) + 0.15 + 0.1 = 1

P(s) = 1 - 0.85

P(s) = 0.15

So the table looks like this after P(s) = 0.15

x	p	q	r	s	t	u
P(x)	0.15	0.25	0.2	0.15	0.15	0.1

Question (b)

P(S) = P(p) + P(q) +P(r) + P(s) +P(t) +P(u)

= 0.15 + 0.25 + 0.2 + 0.15 + 0.15 + 0.1

= 1

P(A) = P(p) + P(q) + P(s) +P(t)

= 0.15 + 0.25 + 0.15 + 0.15

= 0.7

P(B) = P(r) + P(s) +P(t) +P(u)

= 0.2 + 0.15 + 0.15 + 0.1

= 0.6

P(AB) = P(s) +P(t)

= 0.15 + 0.15

= 0.3

Question (c)

A and B are independent if P(A) * P(B) = P(AB)

Here P(A) * P(B) = 0.7 * 0.6 = 0.42

P(AB) = 0.3

P(A) * P(B) P(AB)

Hence A and B are not independent

Question (d)

A and B are mutually exlcusive if P(AB) = 0

Here P(AB) = 0.3

So A and B are not mutually exlcusive

Question (e)

x	p	q	r	s	t	u
P(x)	0.15	0.25	0.2	0.15	0.15	0.1

The table represents the probability distribution of random variable S here.

Since the sum of the probabilities of all the outcomes of S is 1

If the sum of the probabilities of all the outcomes is 1, then it is considered to be a random variable,

S is the random vairable here