Question

Assignment Score: 81.5% Resources [ Give Up? Hint Check Answer KA TY Question 16 of 17 > Suppose Gabe, an elementary school student, has just finished dinner with his mother, Judy. Eyeing the nearby cookie jar, Gabe asks his mother if he can have a cookie for dessert. She tells Gabe that she needs to check his backpack to make sure that he finished his homework. Gabe cannot remember where he left his backpack, but he knows for sure that he did not complete his homework and will not be allowed to eat a cookie. Gabe believes his only option is to quickly steal a cookie while his mother is out of the room. Judy then leaves the room to look for Gabe's backpack. Assume that Judy could return at any time in the next 50 seconds with equal probability. For the first 10 seconds, Gabe sheepishly wonders if he will get caught trying to grab a nearby cookie. After waiting and not seeing his mother, Gabe decides that he needs a cookie and begins to take one from the jar. Assuming it takes Gabe 20 seconds to grab a cookie from the jar and devour it without a trace, what is the probability that his mother returns in time to catch Gabe stealing a cookie? Please round your answer to the nearest two decimal places. P(Gabe gets caught) = about us careers privacy policy terms of use contact us help

Answer 1

Judy can return at any time in 50 seconds after she exits the room. The probability of her returning at any time is the same.

Now, Gabe does not do anything apart from wondering for the first 10 seconds after Judy steps out of the room. And Gabe takes 20 seconds to grab a cookie and devour it with no trace. Now, for Judy to catch Gabe, she should return back within these 20 seconds.

Hence, the probability of Gabe being caught is the same as the probability that Judy returns back within the 20 seconds that Gabe is stealing and devouring the cookie.

Let T be the random variable that describes the time taken by Judy to return back. Since the probability of her returning is the same between the time interval a = 0 seconds and b = 50 seconds, T follows a uniform distribution U(a, b) = U(0, 50).

For uniform distribution, the pdf is given as:

f(:2) » - 9

Substituting the values, we have:

$f(t) = \frac{1}{50 - 0} = \frac{1}{50}$

Now, the time between which Gabe can be caught is 10 and 30 seconds. Hence, we are to determine the probability that T lies in the interval 10 and 30 seconds.

$Pr(10\leq T\leq 30)$

This is given as:

$\\Pr(10\leq T\leq 30) = (30 - 10)\times \frac{1}{50}\\\\ \therefore Pr(10\leq T\leq 30) = 20\times \frac{1}{50}\\\\ \therefore Pr(10\leq T\leq 30) = 0.4$

Hence, the probability of Gabe being caught is 0.40.