The following table shows how a sample of 100 students are distributed in terms of hair and eye color (natural color, not dyed or tinted). Light Hair Dark Hair Light Eyes 25 10 Dark Eyes 15 50 Your friend says, "Hair color and eye color are independent. Just because you have dark hair doesn't automatically mean you have dark eyes." Your friend is wrong, they are not independent. Help explain it to them.
1. The following table shows how a sample of 100 students are distributed in terms of hair and eye color (natural color, not dyed or tinted).
| Light Hair | Dark Hair | |
| Light Eyes | 25 | 10 |
| Dark Eyes | 15 | 50 |
Your friend says, "Hair color and eye color are independent. Just because you have dark hair doesn't automatically mean you have dark eyes." Your friend is wrong, they are not independent. Help explain it to them.
2. Suppose that you are working at an insurance company that uses probability models to help price its life insurance policies. One piece of information that goes into the pricing model is whether or not the person uses tobacco. If the company knows the person uses tobacco, then they charge a higher amount for the life insurance policy, based on the claim that using tobacco and cancer are not independent
3. Explain why, if two events with non-zero probability are mutually exclusive, then they cannot be independent.
4. This one is not really connected to independence so much, but I like it. (You have to answer both parts for this one.)
a. Suppose you know that a friend has 2 children and you know that the older child is a girl. You don't know the sex of the younger child. Explain why the probability that they have two girls is 0.5.
b. Suppose that you have a different friend with 2 children and you know that they have at least one girl. What is the probability that they have two girls? (Hint: it's not 0.5)
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