Dorothy is at a cross roads and needs to decide whether to take the road to the

ID: 3231708 • Letter: D

Question

Dorothy is at a cross roads and needs to decide whether to take the road to the right or the road to the left. She really wants to go to Oz, but she is not sure which road to take. Specifically, she thinks the chances that the road on the right is the road to Oz is 1/2, and that the chances the road on the left is the road to Oz is also 1/2. There is a Scarecrow in the field near the crossroads whom Dorothy can ask directions from. If the Scarecrow points to the right, then Dorothy will think it is twice as likely that the road to the right is the one that goes to Oz than that it is not. If the Scarecrow points to the left, then Dorothy will think it is three times as likely that the road on the left is the one that goes to Oz than that it is not. What are the likelihoods, that is, the probability that the scarecrow points right and that he points left, conditional on the road to the right being the road to Oz, and, similarly, the probability that the scarecrow points right and that he points left, conditional on the road to the left being the road to Oz?

Explanation / Answer

We say that a road is correct if it leads to Oz and it is incorrect otherwise.

Suppose, R = event that road on the RIGHT is correct,

L = event that road on the LEFT is correct,

SR = event that SCARECROW points to the RIGHT and

SL = event that SCARECROW points to the LEFT and

Then, we need to find:

(a) What is the probability that Scarecrow points to RIGHT given that road to RIGHT is correct?

(b) What is the probability that Scarecrow points to LEFT given that road to RIGHT is correct?

(d) What is the probability that Scarecrow points to LEFT given that road to LEFT is correct?

This translates to finding the following values:

(a) P(SR | R) = ?

(b) P(SL | R) = ?

(d) P(SL | L) = ?

Now, as per the question:

P(R) = 1/2 = P(L)

Also, as per question:

P(R | SR) = 2* ( 1 - P(R|SR) ) and P(L|SL) = 3* ( 1 - P(L|SL) )

Solving these, we get:

3 * P(R|SR) = 2 and 4 * P(L|SL) = 3, which gives:

P(R|SR) = 2/3 and P(L|SL) = 3/4

So, we find (a) as:

P(SR | R) = P ( SR and R ) / P(R)

= P(R|SR) * P(SR) / P( (R and SR) or (R and SL) )

= P(R|SR) * P(SR) / ( P(R|SR)*P(SR) + P(R|SL))*P(SL) )

= ( (2/3)*(1/2) ) / ( (2/3)*(1/2) + (1- P(L/SL)) * (1/2) )

= ( (2/3)*(1/2) ) / ( (2/3)*(1/2) + ( (1/4)* (1/2) )

= (2/3) / ( 2/3 + 1/4) = (2/3) * 12 / 11

= 8/11

Now, for (b) P(SL|R) = 1 - P(SR|R) = 1 - 8/11 = 3/11

Similarly, for (c), we get:

P(SR | L) = P ( SR and L ) / P(L)

= P(L|SR) * P(SR) / P( (L and SR) or (L and SL) )

= P(L|SR) * P(SR) / ( P(L|SR)*P(SR) + P(L|SL))*P(SL) )

= ( (1 - 2/3)*(1/2) ) / ( (1 - 2/3)*(1/2) + (3/4) * (1/2) )

= (1/3) / ( 1/3 + 3/4) = (1/3) * 12 / 7

= 4/7

And for (d), we get:

P(SL|L) = 1 - P(SR|L) = 1 - 4/7 = 3/7

This completes the answers.!!

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