4.4 - More Examples

4.4 - More Examples

Example 4-6

A drawer contains:

• 4 red socks
• 6 brown socks
• 8 green socks

A man is getting dressed one morning and barely awake when he randomly selects 2 socks from the drawer (without replacement, of course). What is the probability that both of the socks he selects are green given that they are the same color? If we define four events as such:

• Let $$R_i$$ = the event the man selects a red sock on selection $$i$$ for $$i = 1, 2$$
• Let $$B_i$$ = the event the man selects a brown sock on selection $$i$$ for $$i = 1, 2$$
• Let $$G_i$$ = the event the man selects a green sock on selection $$i$$ for $$i = 1, 2$$
• Let $$S$$ = the event that the 2 socks selected are the same color

then we are looking for the following conditional probability:

$$P(G_1\text{ and }G_2|S)$$

Let's give it a go.

Example 4-7

Medical records reveal that of the 937 men who died in a particular region in 1999:

• 212 of the men died of causes related to heart disease,
• 312 of the men had at least one parent with heart disease

Of the 312 men with at least one parent with heart disease, 102 died of causes related to heart disease. Using this information, if we randomly select a man from the region, what is the probability that he dies of causes related to heart disease given that neither of his parents died from heart disease? If we define two events as such:

• Let $$H$$ = the event that at least one of the parents of a randomly selected man died of causes related to heart disease
• Let $$D$$= the event that a randomly selected man died of causes related to heart disease

then we are looking for the following conditional probability:

$$P(D|H^\prime)$$

The following viewlet uses a Venn diagram to help us work through this problem. Just click on the Inspect! icon when you're good and ready (you'll no doubt want to use the pause and play buttons freely):

If a Venn diagram doesn't do it for you, perhaps an alternative way will:

 [1] Link ↥ Has Tooltip/Popover Toggleable Visibility