Question

P(A) = 0.3, P(B) = 0.7

a. Can you compute P(A and B) if you only know P(A) and P(B)?
b. Assuming that events A and B arise from independent random processes,

i. What is P(A and B)?
ii. What is P(A or B)?
iii. What is P(A|B)?
c. If we are given that P(A and B) = 0.1, are the random variables giving rise to events A and B independent?
d. If we are given that P(A and B) = 0.1, what is P(A|B)?

Answer 1

Answer:

Step-by-step explanation:

Given that

$P(A) = 0.3, P(B) = 0.7$

a) this is not sufficient to calculate P(A and B) unless we know how many entries are common between them

b) Assuming that events A and B arise from independent random processes,

When A and B are independent joint probability would be the product of probabilities

i.  P(A and B)? = $P(A)*P(B) = 0.21$

ii.  P(A or B)=$P(A)+P(B)-P(AB)\\= 0.3+0.7-0.21\\= 0.79$

iii.  P(A|B) = P(A) when A and B are independent.

c. If we are given that P(A and B) = 0.1, are the random variables giving rise to events A and B independent?

No here P(AB) not equals P(A) P(B)So A and B cannot be independent.

d. If we are given that P(A and B) = 0.1,

P(A|B)=$\frac{P(AB)}{P(B)} \\=\frac{0.1}{0.7} \\=\frac{1}{7}$