Conditional Probability ③ Expected Value and Decision-Making
Example ① (Fairness of a Game)
If 1 appears on a die, you receive 500 yen; if 2 or more appears, you pay 100 yen. Expected value:
E=500×(1/6)+(−100)×(5/6)=500/6−500/6=0 yen (fair)
If 1 appears on a die, you receive 500 yen; if 2 or more appears, you pay 100 yen. Expected value:
E=500×(1/6)+(−100)×(5/6)=500/6−500/6=0 yen (fair)
Example ② (Law of Large Numbers)
When a coin is tossed n times, the proportion of heads approaches 1/2 as n→∞. This is the law of large numbers.
When a coin is tossed n times, the proportion of heads approaches 1/2 as n→∞. This is the law of large numbers.
Practice Problems
- Among 100 lottery tickets, 10 are winners with a prize of 1000 yen. Is paying 80 yen for a ticket profitable or not? (Judge using expected value.)
- With two dice, you receive 1000 yen if the product is at least 12, and pay 200 yen otherwise. What is the expected value?
- Find the expected value and variance of X∼B(5, 1/3).
Answers
- Expected value=1000×0.1=100 yen > 80 yen → profitable based on expected value
- Outcomes with product at least 12: (2,6)(3,4)(3,5)(3,6)(4,3)(4,4)(4,5)(4,6)(5,3)(5,4)(5,5)(5,6)(6,2)(6,3)(6,4)(6,5)(6,6)=17 outcomes. E=1000×17/36+(−200)×19/36=(17000−3800)/36≒369 yen
- E(X)=5/3、V(X)=10/9