Probabilistic Livestock Testing

We know the animal selected is actually a cow or a sheep. So, we split this into two cases. Case 1: It is actually a cow. There are 5 cows out of 41 animals, and a 90% chance that the cow will be tested positive as a cow, so the probability that a cow chosen and is tested positive is $\frac {5}{41} \cdot 0.9 =\frac {4.5}{41}$

Case 2: It is actually a sheep. There are 36 sheep out of 41 animals, and a 5% chance it gets tested positive as a cow, so this probability is $\frac {36}{41} \cdot 0.05 = \frac{1.8}{41}$

Thus, the probability it is actually a cow is $\frac {\frac {4.5}{41}}{\frac {4.5}{41} + \frac{1.8}{41}}=5/7$ , so our answer is 5+7=12

There are two ways for the machine to think that the animal is a cow: The animal is a cow and the machine is correct. This occurs with probability of $\frac{5}{41} \times 90\%$ The animal is a sheep and the machine is incorrect. This occurs with probability of $\frac{36}{41} \times 5\%$

The above calculations sum together cover every case where the machine thinks that it is a cow. Clearly, we need the first one to be correct, so that part is the numerator.

Our computed probability is

$\frac{\frac{5}{41} \times 90\%}{\frac{5}{41} \times 90\%+\frac{36}{41} \times 5\%}$ $=\frac{5}{7}$

Our answer is $5+7=\boxed{12}$

total number of animals = 5(cows) + 36(sheep) =41

probability of machine claiming the animal to be cow = (5/41) (90/100) + (36/41) (5/100)

probability of it actually being cow = {(5/41) (90/100)} / {(5/41) *(90/100) + (36/41) (5/100)}

on simplifying it gives = 90/126 = 5/7

therefore answer = 12

There are two cases here. first is that a cow walks in and its condition is 5 and the machine predicts it correctly, Second case is that a sheep walks in and the machine predicts it wrong. The two cases constitute the universal set for this conditional case. The case favoring us is that a cow walks in and the machine predicts it correctly. so the probability is (5 0.9)/(5 0.9+36*0.05 ) = 45/63 =5/7 in the reduced form. so a=5 and b=7. a+b=5+7=12.

Probability of a cow =5/41

Probablity of a sheep=36/41

Probability of being a cow by machine is =(5/41)*9/10+x

x= =Probability of being not a sheep by machine that is actually a sheep=(36/41)*5/100

Therefore,Probability of being a cow by machine is =(5/41) 9/10+(36/41) 5/100

Probability that it is actually a cow =((5/41) (9/10))/((36/41) 5/100+(5/41)*9/10)

Hence answer =5/7 a=5 b=7 a+b=12

Let $A$ be the event that a cow walks through the machine, $A^c$ be the event that a sheep walks through the machine, $B$ be the event that the machine claims that an animal is a cow, and $B^c$ be the event that the machine claims that an animal is a sheep. The desired probability is $P(A|B)$ .

$P(A|B)=\frac{P(A \cap B)}{P(B)}$

$P(A|B)=\frac{P(B|A)P(A)}{P(B|A)P(A)+P(B|A^c)P(A^c)}$

$P(A|B)=\displaystyle\frac{0.9\cdot\frac{5}{5+36}}{0.9\cdot\frac{5}{5+36}+0.05\cdot\frac{36}{5+36}}$

$P(A|B)=\frac{5}{7}$

Thus, $a+b=12$ .

Solution 1:

The probability that a cow walks through the machine is $\frac{5}{41}$ . So the probability that the machine says cow is $0.9\cdot \frac{5}{41} + 0.05\cdot \frac{36}{41} = \frac{0.9(5) + 0.05(36)}{41}$ . The chance that a cow walks through the machine and the machine says cow is $0.9\cdot \frac{5}{41}$ , so the probability that it is actually a cow is $\frac{\frac{.9(5)}{41}}{\frac{.9(5)+.05(36)}{41}} = \frac{.9(5)}{.9(5)+.05(36)} = \frac{5}{7}.$ So $a + b = 5 + 7 = 12$ .

Solution 2: This is a simple application of Bayes Rule. Let $A$ be the event that a cow walks through the machine. Let $B$ be the event that the machine claims the animal is a cow. $P(A) = \frac{5}{41}$ and $P(B) = \frac{5}{41}\cdot \frac{9}{10} + \frac{36}{41}\cdot \frac{1}{20} = \frac{126}{820} = \frac{63}{410}$ and $P(B|A) = \frac{9}{10}$ . So $P(A|B) = \frac{P(B|A)P(A)}{P(B)} = \frac{9}{10}\cdot \frac{5}{41}\cdot \frac{410}{63} = \frac{45}{63} = \frac{5}{7}$ . So $a + b = 5 + 7 = 12$ .

Use simple Bayes theorem

I drew a table comparing what the animals are and what the machine claims them to be. Since it claims it's a cow, we see that 1.8 sheep would be tested as a cow and 4.5 cows would be tested as a cow. Therefore, 4.5 / 1.8 + 4.5 = 5/7.

a/b=45/(45+18)=45/83=5/7..............a+b=12