Question

IV. Consider the rype of clotbes dryer (gas or electric) purchased by each of five different customers at a certain store. 1. If the probability that at mo what is the probability tbat at least two purchase an electric dryer 2 lf P(all fve purchase gas)116 and P(all five parchase electric-005.wha: is the probability that at least one of each tpe is purchased st one of these purchases an electric dryer is 428. V. Two pumps connected in parallel fail indepeadently of ope anotber oc & gver day The probability tha: only the oider pump ill fal is 10. and the probability that only the pewe: pump will fail is 0.05. Wha: is the probabiliry that the pumping system will fail on any given dav (which happens if both pumps fail s just taken delivery or a lor of 10.0002 x4 boards Sup- VI A lumber company has pose tha: 20% of these boards (2.000) are actualy too green to be used in firs: quality construction. Two boards are seiected one after the ocher Let A=the firs: is greer and Athe second is green 1Compute P(A). P(B). and PlAD B). Are A and B independent? 2 With A and B independen: and P(A)= PB) = 2 what is PLAr B) How much difference is there berweet this answer and PAnB it part (1)? 3. Suppose the lot consists of ten boards. of whch two are green Does the assumption of independence now yicid approximately the correct answer for P(AnB)What is the critical difference between the situation here and that of part (1)? VIL. A vaccine has 90% probability of being effective in preventing & certain disease The probabilitry of gerting the disease if a person is not vaccinated is 50% In & certain geographic region. 25% of the peopie get vaccinated. If a person is seiected a: random. find the probabiliry that he or she will contract the disease VII. Two bowls each contain 5 black and 5 white balls A ball is chosen at random from bowl 1 and put into bowl 2. A ball is ther choser at random from bowl 2 and put into bowl 1. Find the probability that bowl 1 still bas 5 black and 5 white balls

Answer 1

Answer:

V)

Probabilit of at laast tro punchase an eleetrie dryep = 1- P(at most one of these austomays Punchases an electrie drygp. E 0.572 PC au five purchase gas) o 16 = PC) ,(say ) P(au fve puchase electrie) - o.005PCB), (say) of Rach type is punchasid) Plat least onL [P)P) +t 0.121 ニ 0.879

VI)

Let A be the event that the older pump will fail Let B be the event that the newer pump will fail Let An Bbe the event that both the pumps fail. The event that only older pump will fail is shown in Venn diagram as follows: Only A B A The probability of this event is given by: P(Only A) P(A)-P(AnB) Given that this value is 0.10 So, P(Only A) P(A)-P(AnB) 30.10-P(A)-P(AΩΒ) P(A)-P(AnB)= 0.10 P(A) 0.10+ P(AnB)

The event that only newer pump will fail is shown as shaded region in the following Venn diagram: Only B B A

The probability of this event is given by: P(Only B) P(B)-P(AB) Given that this value is 0.05. So, P(Only B) P(B)-P(AnB) 0.05 P(B)-P(AnB) P(B)-P(An B) = 0.05 P(B) 0.05+P(AnB)

The pumping system will fail on any given day if both pumps fail. The event that both pumps fail is equal to the event AnB Since the two pumps fail independently of one another, the probability of the event An B is given by: P(AnB) P(A) P(B) -(0.10+P(AnB)(0.o5+ P(AnB)

Let P(An B) x So, it can be rewrite as: P(AnB)=(0.10+P(AnB))x(0.05+P(AnB)) (0.10+x)(0.05+x) x0.005+0.10x+0.05x +x x0.005+0.15x+x 0.005-0.85x + x2 = 0 (x-0.8441) (x-0.0059)= 0 x -0.8441 or x 0.0059 Since the probability cannot be greater than the either of 0.05 or 0.10. To satisfy this condition the required probability value is P(AnB) = 0.0059|