Question

Many utility companies promote energy conservation by offering discount rates to con-sumers who keep their energy usage below certain established subsidy standards. Arecent report notes that 70% of the people live in Phnom Penh have reduced their elec-tricity usage sufficiently to qualify for discounted rates. If five residential subscribers are randomly selected from Phnom Penh, find the probability of each of the following events:

(a) All five qualify for the favorable rates.

(b) At least four qualify for the favorable rates.

(c) At least two do not qualify the favorable rates.

 

Answer 1

Solution

(a) All five qualify for the favorable rates.

Let X be the number of residential subsriber who qualify for thhe favorable rates :

\( P(qualify)=0.7,\hspace{2mm} \) Then \( \hspace{2mm} X\sim Bin(5,0.7) \)

\( \implies P(X=5)=(0.7)^5=0.16807 \)

Therefore.  \( P(X=5)=0.16807 \)

(b) At least four qualify for the favorable rates.

\( \implies P(X\geq 4)=P(X=4)+P(X=5) \)

                            \( =5\times (0.7)^4\times (0.3)+16807 \)

                            \( =0.52822 \)

Therefore.  \( P(X\geq 4)=0.52822 \)

(c) At least two do not qualify the favorable rates.

\( \implies P(5-X\geq 2)=P(X\leq 3)=P(0)+P(1)+P(2)+P(3) \)

                                   \( =0.78048 \)

Therefore. \( P(X\leq 3)=0.78048 \)

Therefore.

a). \( P(X=5)=0.16807 \)

b). \( P(X\geq 4)=0.52822 \)

c). \( P(X\leq 3)=0.78048 \)