Which of the following statements are true? Mark all that apply.
If the Expected Value of a game is 1, we say that it is a 'fair game'
If the odds for an event are 3 to 5, then the odds against that event are 5 to 3.
Expected Value is the actual gain or loss each time an event takes place.
The probability of an impossible event is 0.
Expected Value is the average gain or loss of an event if the procedure is repeated many times.
The probability of an event that is certain is 1.
To determine which of the statements are true, let's analyze each one:
1. If the Expected Value of a game is 1, we say that it is a 'fair game': True
Explanation: A game is considered fair if the expected value is equal to 1. The expected value represents the average outcome when playing the game repeatedly, and a value of 1 indicates that, on average, the player neither gains nor loses anything.
2. If the odds for an event are 3 to 5, then the odds against that event are 5 to 3: True
Explanation: When odds in favor of an event are given, they can be expressed as a ratio. To determine the odds against the event, you simply swap the numerator and denominator. So, if the odds for an event are 3 to 5, the odds against that event would be 5 to 3.
3. Expected Value is the actual gain or loss each time an event takes place: False
Explanation: The expected value is not the actual gain or loss of a single event but rather the average gain or loss when the event is repeated many times.
4. The probability of an impossible event is 0: True
Explanation: An impossible event has no chance of occurring, so its probability is indeed 0. Probability ranges from 0 (impossible) to 1 (certain).
5. Expected Value is the average gain or loss of an event if the procedure is repeated many times: True
Explanation: As mentioned earlier, the expected value represents the average gain or loss when an event is repeated numerous times.
6. The probability of an event that is certain is 1: True
Explanation: If an event is certain, it means it is guaranteed to occur. In such a case, the probability of the event is 1, indicating a 100% chance of occurrence.
In summary, the true statements are:
- If the Expected Value of a game is 1, we say that it is a 'fair game'
- If the odds for an event are 3 to 5, then the odds against that event are 5 to 3.
- The probability of an impossible event is 0.
- Expected Value is the average gain or loss of an event if the procedure is repeated many times.
- The probability of an event that is certain is 1.