Question

Uptown, 60% of residents live in apartments and 40% of residents recycle. The probability of living in an apartment or recycling is 80%.

(a) What are the events in this question?

(b) In event notation, what probabilities were you given in the question?

(c) What is the probability that a randomly selected resident both lives in an apartment and recycles?

(d) If the person lives in an apartment, what is the probability that they recycle?

(e) For a randomly selected resident, are living in an apartment and recycling independent? Briefly justify your answer.

Answer 1

a)

i) Event that residents live in apartments

ii) Event that residents recycle.

b)

Let residents live in apartment be A and residents recycle be R then,

P(A) = 0.60, P(R) = 0.40

Given also, P(A OR R) = 0.80

c)

Using addition rule,

P(A or R) = P(A) + P(R) - P( A and R)

P(A and R) = P(A) + P(R) - P(A or R)

= 0.60 + 0.40 - 0.80

= 0.20

d)

Using conditional rule,

P(R | A ) = P( A and R) / P(A)

= 0.20 / 0.60

= 1/3

= 0.3333

e)

Two events are independent if P(A and R) = P(A) * P(R)

Now P(A and R) = 0.20

P(A) * P(R) = 0.60 * 0.40 = 0.24

Since P(A and R) P(A) * P(R) ,

The events living in apartment and recycling are not independent.