Dogs are inbred for such desirable characteristics as blue eye color, but an unfortunate by-product of such inbreeding can be the emergence of characteristics such as deafness. A 1992 study of Dalmatians (by Strain and others, as reported in The Dalmatians Dilemma) found the following:(i) 31% of all Dalmatians have blue eyes.(ii) 38% of all Dalmatians are deaf.(iii) If a Dalmatian has blue eyes, there is a 42% chance that it is deaf.Based on the results of this study is "having blue eyes" independent of "being deaf"? No, since 0.38 is not equal to 0.42. Yes, since 0.38 is not equal to 0.42. No, since 0.31 is not equal to 0.42. Yes, since 0.31 * 0.38 is not equal to 0.42.
Question
Dogs are inbred for such desirable characteristics as blue eye color, but an unfortunate by-product of such inbreeding can be the emergence of characteristics such as deafness. A 1992 study of Dalmatians (by Strain and others, as reported in The Dalmatians Dilemma) found the following:(i) 31% of all Dalmatians have blue eyes.(ii) 38% of all Dalmatians are deaf.(iii) If a Dalmatian has blue eyes, there is a 42% chance that it is deaf.Based on the results of this study is "having blue eyes" independent of "being deaf"? No, since 0.38 is not equal to 0.42. Yes, since 0.38 is not equal to 0.42. No, since 0.31 is not equal to 0.42. Yes, since 0.31 * 0.38 is not equal to 0.42.
Solution
The question is asking whether the events "having blue eyes" and "being deaf" are independent. Two events are independent if the probability of one event occurring does not affect the probability of the other event occurring.
In this case, we are given that:
(i) P(Blue Eyes) = 0.31 (ii) P(Deaf) = 0.38 (iii) P(Deaf | Blue Eyes) = 0.42
If "having blue eyes" and "being deaf" were independent, then P(Deaf | Blue Eyes) would be equal to P(Deaf), which is 0.38. However, we are given that P(Deaf | Blue Eyes) = 0.42, which is not equal to 0.38.
Therefore, "having blue eyes" is not independent of "being deaf". So, the correct answer is "No, since 0.38 is not equal to 0.42."
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