Solving a Brilliant Math Problem

The probability that Katarina will correctly solve a given Brilliant math problem is $\frac {1}{8}$ . The probability that Layla will solve the same problem correctly is $\frac {1}{12}$ , therefore the probability that both of them give correct answer is $\frac {1}{8} \times \frac {1}{12} = \frac {1}{96}$ .

The probablility that Katarina will give incorrect solution is $1-\frac {1}{8}=\frac {7}{8}$ & The probablility that Layla will give incorrect solution is $1-\frac {1}{12}=\frac {11}{12}$ , therefore the probability that both of them give incorrect answer is $\frac {7}{8} \times \frac {11}{12} = \frac {77}{96}$ . Also it is given that the probability that they will give the same incorrect numerical result is $\frac {1}{1001}$ . So the probability that they get incorrect result as well as same incorrec result is $\frac {1}{1001} \times \frac {77}{96} = \frac {1}{1248}$ .

Total probability that they get same result = getting same incorrect result + getting same correct result. i.e. $= \frac {1}{1248} + \frac {1}{96} = \frac {7}{624}$ therefore probability of getting of both getting correct answer if given that both have got same answer $= \frac {1}{96} / \frac {7}{624}$ Simplifying it gives $\frac {13}{14}$ . As 13 & 14 are both coprime so they could be expressed in form of $\frac {a}{b}$ . $\Rightarrow$ $a=13, b=14, a+b = 13+14=27$ .

[latex edits]

We know the probability when they both get the question correct is 1/8 * 1/12 = 1/96.

The probability that they both get it incorrect is 7/8 * 11/12 = 77/96.

We know that if they both got it incorrect, the probability of them getting the same answer is 1/1001: hence the probability of the two getting the SAME answer is:

1/96 + 77/96 * 1/1001.

Then the probability of getting the question correct is \frac{1/96}{1/96+77/96*1/1001} = 13/14 implying 27.

The probability that both are incorrect is 7/8 x 11/12 and the probability they are both incorrect and gave the same answer is 7/8 x11/12 x 1/1001= 1/(96 X 13).

The probability that they are both correct is 1/8 x 1/12= 1/96.

Now it is time for Bayes : Probability (Both Correct | Same answer) = Probability (SA |Both Correct) / Prob (Same answer) or

Probability (Both Correct | Same answer) = 1/96 /(1/96+1/(96 x 13))=13/14

The probability of both of the girls solving the question correctly is 1/96 and probability of both of the girls solving the question incorrectly is 77/96 .. Now let's consider 96096(96*1001) cases out of which 77077 cases would be the cases in which the girls have solved the question incorrectly.. Given that the probability of having same incorrect numerical result is 1/1001 when both of them had solved it wrong. Therefore in 77077 incorrect cases ,there would be 77 cases in which the girls will be having same incorrect value. Out of 96096 cases , 1001 cases are the ones solved by both the girls correctly as probability is 1/96. So total no. Of cases which have same numerical value =1001+77=1078 .

Hence ,probability of getting a correct answer when they both got same numerical result =1001/1078 .

1001 and 1078 have common factors as 7 and 11. So final probability comes to be 13/14. The final answer is 13+14 =27 .

Let \­(BW\­) be the event that Katarina and Layla are both wrong, $BC$ be the event that they are both correct and $S$ be the event that they both get the same answer. We have:

• $P(BC)=\frac {1\times1}{8\times12}=\frac{1}{96}$

• $P(BW)=\frac{7\times11}{8\times12}= \frac{77}{96}$

• $P(S|BW)=\frac{1}{1001}$

Now $P(S|BW)\timesP(BW)=P(S\capBW)=\frac{77}{96096}$ .

And of course, $P(BW\cupS)=P(BW)+P(S)-P(BW\capS)$ =P(S)+\frac{77}{96}-\frac{77}{96096}).

If K and L are not both wrong then at least one must be right, and if they also cannot have the same answer then exactly one must be right. Thus $P(BW\cupS)=1-P(exactly one girl is correct)=1-\frac{11+7}{96}=\frac{78}{96})$ .

Now $P(S)=\frac{78}{96}+\frac{77}{96096}-\frac{77}{96}=\frac{1078}{96096}$ .

Now we want to find $P(BC|S)$ . But if $S$ is true, we must have either $BC$ or $BW$ .

So $P(BC|S)=1-P(BW|S)=1-\frac{P(BW\capS)}{P(S)}=1-\frac{77}{1078}=\frac{1001}{1078}=\frac{13}{14}$ .

Finally, $13+14=27$ .

they solved independently so the result can be correct or wrong the probability that the answers correct and they got same value is both getting correct answer i.e 1/8x1/12=1/96 the probability that the answers wrong and they got same value is 7/8x11/12x1/1001=1/(96x13) so the probability that both get correct when given that they get same numerical will be (1/96)/(1/96+1/(96x13))=13/14 so a+b=27

Using Bayes' theorem one can see that P(both are correct | both got the same answer) = P(both got the same answer | both are correct) * P(both are correct) / P(both got the same answer).

We use

P(both got the same answer | both are correct) = 1

P(both are correct) = 1/8 * 1/12

P(both got the same answer) = 1/8 * 1/12 + 7/8 * 11/12 * 1/1001

where the last line says that the probability they got the same answer is the probability they're both correct, plus the probability they're both wrong yet have the same numerical value.

Let $C$ be the event that Katarina and Layla both get the correct result. Let $S$ denote the event that Katarina and Layla get the same result. Let $I$ denote the event that Katarina and Layla get the same incorrect result.

We want to find the conditional probability $P(C|S)$ . By definition, this is equal to $P(C \cap S) / P(S)$ . But $P(C \cap S) = P(C)$ , and $P(S) = P(C) + P(I)$ . From the information given, we get $P(C) = \frac{1}{8} \times \frac{1}{12}$ , and $P(I) = \frac{7}{8} \times \frac{11}{12} \times \frac{1}{1001}$ . Therefore, $P(C|S) = ( \frac{1}{8} \times \frac{1}{12} ) / ( \frac{1}{8} \times \frac{1}{12} + \frac{7}{8} \times \frac{11}{12} \times \frac{1}{1001} ) = \frac {1001}{1001+77}$ . This expression simplifies to $\frac{13}{14}$ , and so $a+b = 13+14 =$ 27.