Naruto is choosing between two venues that will deliver food to his house. With probability 1/3 he will choose Ramen Mania (R), and with probability 2/3 he will choose Sushi (S). If he chooses R, 15 minutes after making the call, the remaining time it takes the food to arrive is exponentially distributed with average 10 minutes. If he orders S, 10 minutes after making the call, the time it takes the food to arrive is exponentially distributed with an average of 12 minutes. Given that he has already waited 25 minutes after calling, and the food has not arrived, what is the probability that he ordered from R?

39.10%

Step-by-step explanation:

The given timeout distributions are:

Exp (1/10), Exp (1/12)

The equation to use is the following:

F (x) = 1 - e ^ ( - a / b); x> 0

So the probability is:

P (food has not arrived after waiting 25 minutes | R) = e ^ ( - 10/10) = 0.3679

P (food has not arrived after waiting 25 minutes | S) = e ^ ( - 12/15) = 0.2865

So:

P (food has not arrived after waiting 25 minutes) = (food has not arrived after waiting 25 minutes | R) * P (R) + (food has not arrived after waiting 25 minutes | S) * P (S)

Replacing:

P (food has not arrived after waiting 25 minutes) = 0.3679 * 1/3 + 0.2865 * 2/3 = 0.3136

However:

P (R | food has not arrived after waiting 25 minutes) = (food has not arrived after waiting 25 minutes | R) * P (R) / P (food has not arrived after waiting 25 minutes)

P (R | food has not arrived after waiting 25 minutes) = 0.3679 * 1/3 / 0.3136 = 0.3910

It means that the probability is 39.10%