Question

Australian cattle are known to have a high prevalence of congenital deafness. Deafness in both ears is referred to as bilateral deafness, while deafness in one ear is referred to as unilateral deafness.Cattle have a "mask" around one or both eyes (in other words, a unilateral mask or a bilateral mask). Suppose that 40% of Australian cattle have black coats; these individuals are commonly referred to as "Blue Heelers" as opposed to "Red Heelers". Among Blue Heelers, 35% of individuals have bilateral masks and 25% have unilateral masks. About 50% of Red Heelers exhibit no eye masking and 10% have bilateral masks. Let M represent the event that an Australian cattle has a facial mask, where M2 represents a bilateral mask, M1 represents a unilateral mask, and M0 indicates lack of a mask.a) Calculate the probability an Australian cattle has a facial mask and a black coat.b) Calculate the prevalence of bilateral masks in Australian cattle.c) Among Australian cattle with bilateral facial masks, what is the probability of being a Red Heeler?d) Unilateral deafness occurs in Red Heelers with probability 0.15, in both dogs that either lack facial masking or exhibit a unilateral mask; for both unmasked and unilaterally masked Red Heelers, 60% of dogs are not deaf. The overall prevalence of bilaterally masked Australian cattle with bilateral deafness and red coats is 1.2% and the overall prevalence of bilaterally masked Australian cattle with unilateral deafness and red coats is 4.5%; these prevalences are the same for Australian cattle with black coats. Among Blue Heelers with either no facial masking or a unilateral mask, the probability of unilateral deafness is 0.05 and the probability of bilateral deafness is 0.01. Let D represent the event that an Australian cattle is deaf (i.e., deaf in at least one ear), where D2 represents bilateral deafness and D1 represents unilateral deafness.i. What is the probability that an Australian cattle has a bilateral mask, no hearing deficits, and a red coat?ii. Calculate the proportion of bilaterally masked Blue Heelers without hearing deficits.iii. Compare the prevalence of deafness between Red Heelers and Blue Heelersiv. If a dog is known to have no hearing deficits, what is the probability it is a Blue Heeler?

Answer 1

Answer:

Step-by-step explanation:

Answer:

A. 0.096

B. 9.2%

C .0.94

Di 0.943

Dii. 13.16%

Diii 0.05504

Step-by-step explanation:

Given

Blue Heelers

PR(B) = 0.4,

PR(M1) = 25% 0f 40% = 0.1

PR(M2) = 35%'of 40% = 0.14

Red Heelers

PR(Rll = 0.6,

PR(M1) = 40% of 60%=0.24

PR(M2) =10% of 60% = 0.06

a. The required probability is

PR[(M1nB)U(M2nB)] = 0.1 ×0.4 + 0.14 × 0.4

= 0.096

b. The required probability is:

PR[(BnM2)U(RnM2)]

= 0.4 × 0.14 + 0.6 × 0.06

= 0.092

= 9.2%

c.The required probability is:

PR(M2nR) = PR(M2nR) = 0.06 × 0.6

= 0.036

d.I. The required probability is PR[M2nD"nR]

PR(M2nD2nR) =0.012

PR((M2D1nR) =0.045

We can find PR(M2nD'nR)

PR( M2nD"nR) = 1-(0.012 +0.045)

PR(M2nD"nR) = 0.943

dii. For Blue Heelers Given

PR(D1l =0.05

PR(D2) = 0.01

PR(D’) = ?

PR(D') + PR(D1) + PR(D2) = 1

P(D') = 1 - (0.05 + 0.01)

P(D') = 0.94(Blue Heelers)

= 94%

PR( M2nD' ) = PR(M2) × PR(D')

= 0.14 × 0.94

= 0.1316

= 13.16 %

diii

Blue Heelers. Red Heeler

D1. 0.05.

D2. 0.01

D'. 0.94

Div The reqired Probability is

PR[BnM1nD"] + PR[BnM2nD"]

= 0. 4 × 0.14 × .94 + 0.4 ×0.1 × 0.94

= 0.07144 + 0.0376

= 0.05504

Answer 2

Answer:

a) 0.24

b) 70%

c) 0.10

d) i. 0.0012

ii. 40%

iv. 0.96

Step-by-step explanation:

Let black coats (blue heelers), B1 = 40%

Red heeler, R1 = (100 - 40)% = 60%

B2(M2), Blue heelers with bilateral mask = 35%

B2(M1), Blue heelers with unilateral mask = 25%

R2(M0), Red heelers with no mask = 50%

R2(M2), Red heelers with bilateral mask = 10%

a) B2 = B2(M2)  + B1(M1) = (35 + 25) = 60%

P(B2) = 0.60 and P(B1) = 0.40

∴  P(B2 ∩ B1) = P(B2) * P(B1) = 0.60 * 0.40 = 0.24

b) M = M1 + M2 ( M1 = 0.25; M2 = 0.35 + 0.10 = 0.45)

M = 0.25 + 0.45 = 0.70 = 70%

c) R2(M2) = 10%

P(R2(M2) = 0.10            

d) i. P(R1 ∩ D2) = 0.12 X 0.01 = 0.0012

ii. 1 - 0.60 = 0.40 = 40%

iii. blue and red heelers with unilateral deafness = 0.05 and 0.15 respectively

blue and red heelers with bilateral deafness = 0.01 and 0.30 ( i.e. 0.45 - 0.15 = 0.30) respectively

iv.P(B1) = 1 - (0.05 + 0.01) = 1 - 0.06 = 0.96