| Hair | |||||
| Dark | Light | Red | Total | ||
| Blue | 0.03 | 0.15 | 0.02 | 0.2 | |
| Eye | Brown | 0.27 | 0.45 | 0.08 | 0.8 |
| Total | 0.3 | 0.6 | 0.1 | 1 | |
1.1 What is the probability of having red hair, if one has brown eyes? 0.1
1.2 What is the probability of having brown eyes or light hair? 0.95
1.3 What is the probability of having dark hair and brown eyes? 0.27
1.4 What is the correlation between dark hair and blue eyes? Explain. Negative, because 0.03 = P(dark hair and blue eyes) < P(dark hair)*P(blue eyes) = 0.2*0.3=0.06.
2. Construct the Venn diagram giving the probabilities of hair and eye coloration using the following information. It will not have the same numbers as #1. People with brown eyes are 75% of the population. Ten percent have light hair and blue eyes. If a person has blue eyes, then there is a 35% chance they have red hair. Red hair and blue eyes are independent. Half the population has dark hair.
| Hair | |||||
| Dark | Light | Red | Total | ||
| Blue | 0.25-0.1-0.0875 =0.0625 | 0.1 | 0.35*0.25=0.0875 | 1-0.75=0.25 | |
| Eye | Brown | 0.5-0.0625 =0.4375 | 0.15-0.1=0.05 | 0.35-0.0875 =0.2625 | 0.75 |
| Total | 0.5 | 1-.5-.35=0.15 | 0.0875/0.25=0.35 | 1 | |
3. You are searching for a German battleship (WWI). It is in either the North Atlantic Ocean, North Sea, or South Atlantic Ocean. Your first guess for the probabilities of being in these waters are 0.4, 0.25 and 0.35 respectively. In addition the conditional probabilities of detecting the battleship are 0.2, 0.3 and 0.4 respectively.
3.1 What is the overall probability of a detection, if all three are searched? 0.295
3.2 If you look in all three places and do not find the battleship, what is the probability that it is in the North Sea? 0.2482
3.3 If you could look in only one of these places, where would you look? What is the probability of a detection there? South Atlantic Ocean 0.14
3.4 If you do not look in all three places, but only look in the North Sea, what is the probability of a detection? 0.075
3.5 If you look first in only the North Sea and do not find it, where (only one place) should you look the second time? What is the probability of a detection on this search? South Atlantic Ocean 0.15135