Example ① A lottery pays 1,000,000 yen for first prize (probability 0.0001), 10,000 yen for second prize (probability 0.01), and 0 yen otherwise. A ticket costs 300 yen. What is the expected gain from buying one ticket?
Solution:E(X) = 1000000·0.0001 + 10000·0.01 + 0·… = 100 + 100 = 200 yen.
Expected gain = 200 − 300 = −100 yen (buying it results in a loss)
Solution:E(X) = 1000000·0.0001 + 10000·0.01 + 0·… = 100 + 100 = 200 yen.
Expected gain = 200 − 300 = −100 yen (buying it results in a loss)
Key Point
- Expected-value calculations help evaluate the soundness of decisions
Practice Problems
- A game has a winning probability of 1/4 and a prize of 100 yen. The entry fee is 30 yen. What is the expected gain?
Solutions and Explanations
- Solution:−5 yen
100·1/4−30=25−30=−5.