In the popular TV cartoon series, \"The Simpsons,\" the Simpson family has five

Question

In the popular TV cartoon series, "The Simpsons," the Simpson family has five members: Homer, Marge, Bart, Lisa, and Maggie. Let's involve the Simpsons in Simpson's Paradox. Consider Homer and Bart's consumption of potato chips and donuts. Suppose that Marge is in the room 20% of the time that Bart is eating chips and 40% of the time that Bart is eating donuts. Similarly, suppose that Marge is in the room 20% of the time that Homer is eating chips and 40% of the time that Homer is eating donuts; thus she catches them eating each kind of snack equally often. Nonetheless, Marge is in the room 24% of the time that Bart is eating either donuts or chips, and Marge is in the room 34% of the time that Homer is eating either donuts or chips. Suppose that Homer and Bart never eat chips and donuts in the same snack--eating chips and eating donuts are mutually exclusive.

1. What fraction of the time that Bart eats either chips or donuts does he eat donuts?

2. What fraction of the time that Homer eats either chips or donuts does he eat donuts?

Explanation / Answer

Q1

Let proportion of time Bart eats chips = p and proportion of time Bart eats donuts = q.

Given that Homer and Bart never eat chips and donuts in the same snack, fraction time Bart eats chips or donuts = p + q.

Now,

out of this p fraction of times, fraction of time Marge catches Bart eating chips = 0.2 and

out of this q fraction of times, fraction of time Marge catches Bart eating donuts = 0.4. So,

fraction of time Marge catches Bart eating chips or donuts = 0.2p + 0.4q.

This is given to be 0.24.     

So, 0.2p + 0.4q = 0.24(p + q) or 0.16q = 0.04p or p = 4q.

Then,

fraction of time that Bart eats either chips or donuts does he eat donuts = q/( p + q) = q/5q = 1/5 = 0.2 ANSWER

Q2

Let proportion of time Homer eats chips = r and proportion of time Homer eats donuts = s.

Given that Homer and Bart never eat chips and donuts in the same snack, fraction time Homer eats chips or donuts = r + s.

Now,

out of this r fraction of times, fraction of time Marge catches Homer eating chips = 0.2 and

out of this s fraction of times, fraction of time Marge catches Homer eating donuts = 0.4. So,

fraction of time Marge catches Homer eating chips or donuts = 0.2r + 0.4s.

This is given to be 0.34.     

So, 0.2r + 0.4s = 0.34(r + s) or 0.06s = 0.14r or r = (3/7)s.

Then,

fraction of time that Homer eats either chips or donuts does he eat donuts = s/( r + s) = s/{(3/7)s + s} = 7/10 = 0.7 ANSWER

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