Probability – Expected Value Examples Explained

In the Swedish Parking Example that Bob Heckard from Penn State University provides from his Lectures notes a car is parked in a two hour parking zone. A police man comes strolling by and checks the “clock position” of two of the driver’s four tires, because if a car moves the “clock position” of its air valve stems change. The car would be ticketed in two hours if the valves are still in the same positions, implying that the car had not moved: however, there is a possibility that the air valve stem returned to the same place after the car was moved. The probability of the stem being in the same place is (1/12) x (1/12) = 1/144. The probabilities of each individual tires’ air valve stem is in the same place are multiplied because they are conditional. This is expected value because it is the mean over a long period of time: Once in every one hundred and forty four times a car is parked its air valve stems will be in the clock position.
Heckard also presents another example of a family planning strategy. If a family has children until they have a girl or stops after two boys will this change the male-female population? The mean probability for both girls and boy are 0.75. Therefore the sex mix or boy-girls stays equal, and the overall popualation ratio of boys and girls stays equal. This is expected value because the probability is taken over many generations, making the value a long term mean.
Heckard’s last example is a roulette wheel. A roulette wheel has thirty eight numbers: eighteen are red, eighteen are black and two are green. When a player bets if he/she is right they gain thirty five dollars, but if they are wrong the lose one dollar. The win probability is 1/38, and the lose probability is 37/38. The expected value is negative one-half, meaning over many bets the player will lose five cents per dollar bet. A roulette wheel is not far, because the negative out come means that the casino actually benefits from this game. This outcome is expected value because in the calculation the right and wrong bets of money were incorporated into the equation. [(35)(1/38)+(-1)(37/38)] = -0.05 This shows the probability over many bets, which is the expected value.