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Exercise 2.57

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(a)

Suppose

C_{1}

contains

7

green gummi bears and

8

red ones,

M_{1}

contains

1

green gummi bear and

2

red gummi bears,

C_{2}

contains

5

green gummi bears and no red gummi bears,

M_{2}

contains

12

green gummi bears and

5

red gummi bears.

The proportion of green gummi bears in $C_{1}$ is $\frac{7}{15}$ , which is larger than that of $M_{1}$ , which is $\frac{1}{3}$ . The proportion of green gummi bears in $C_{2}$ is $\frac{5}{5}$ , which is larger than that of $M_{2}$ , which is $\frac{12}{17} .$ However, the proportion of green gummi bears in $C_{1} + C_{2}$ is $\frac{12}{20}$ , which is less than that of $M_{1} + M_{2}$ , which is $\frac{13}{20} .$

(b)

We can imagine that it is much more difficult to get a green gummi bear out of a jar with subscript

1

than it is out of a jar with subscript

2

C

jars have a lower overall success rate, because most of their green gummi bears are in

C_{1}

, which is harder to sample from compared to the jars with subscript

2

Let $A$ be the event that a sampled gummi bear is green. Let $B$ be the event that the jar being sampled from is an $M$ jar. Let $C$ be the event that the jar being sampled from has subscript $1$ .

Then, by Simpson’s Paradox, $P (A | B, C) < P (A | B^{c}, C)$ , $P (A | B, C^{c}) < P (A | B^{c}, C^{c})$ , however, $P (A | B) > P (A | B^{c}) .$

fifthist

2021-12-05 00:00

Exercise 2.57

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