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3.3 Probability of Unions
Imagine that you toss a fair, two-sided quarter. You let it land and take a look at the side facing
up. What is the probability that you see heads or tails (assume the toss will be ignored if it
happens to land on its side)
You can probably see fairly quickly that the outcome desired is guaranteed; when a coin is
tossed, it will result in one of two outcomes: heads or tails. If someone in a bet were to tell you
that he will win if the toss of a coin results in heads or tails, then you could probably tell him,
“Congratulations!”
Adding to our intuition (no pun intended), we will write the situation in the form of a
mathematical probability. The sample space will have two outcomes:
Then,
( )
Since we know that
( ) ( )
So, we can gladly write:
( ) ( ) ( )
Simple enough! We feel pretty satisfied and so we hope to
tackle another problem:
Example 1: A large company offers a self-insured health
insurance policy to its employees to help them reduce premium and copay costs. Using its
historical data from the last two years, the company analyst considers the risk status of the
employees (low or high) based on preexisting conditions, and the type of claim filed (physical
health or mental health). He finds that 70% of employees have filed a mental health claim and
that 40% of employees have been categorized as high risk. Further, he finds that 20% of
employees are low risk and have filed a physical health claim. The company only insures the
first claim. All claims thereafter are paid for by a third-party insurer.
Statistics for Decision-Making in Business © Milos Podmanik Page 99