Question

which of the following statements are true? mark all that apply.
the probability of an event that is certain is 0.
expected value is the average gain or loss of an event if the procedure is repeated many times.
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 - 1.
if $p(a)$ is the probability that event $a$ will occur, the probability event $a$ will not occur is $p(overline{a})=1 - p(a)$.
expected value is the actual gain or loss each time an event takes place.
if the expected value of a game is zero, we say that it is a fair game

Explanation:

Step1: Recall probability basics

The probability of a certain - event is 1, not 0. So the statement "The probability of an event that is certain is 0" is false.

Step2: Define expected value

Expected value is the average gain or loss of an event if the procedure is repeated many times, not the actual gain or loss each time. So the statement "Expected Value is the actual gain or loss each time an event takes place" is false, and the statement "Expected Value is the average gain or loss of an event if the procedure is repeated many times" is true.

Step3: Understand odds

If the odds for an event are \(a:b\), then the odds against the event are \(b:a\). So if the odds for an event are 3 to 5, then the odds against that event are 5 to 3, and this statement is true.

Step4: Recall probability of impossible event

The probability of an impossible event is 0, not - 1. So the statement "The probability of an impossible event is - 1" is false.

Step5: Use complementary - probability formula

If \(P(A)\) is the probability that event \(A\) will occur, then the probability that event \(A\) will not occur is \(P(\overline{A})=1 - P(A)\), and this statement is true.

Step6: Recall fair - game definition

If the expected value of a game is zero, we say that it is a 'fair game', and this statement is true.

Answer:

• Expected Value is the average gain or loss of an event if the procedure is repeated many times.
• If the odds for an event are 3 to 5, then the odds against that event are 5 to 3.
• If \(P(A)\) is the probability that event \(A\) will occur, the probability event \(A\) will not occur is \(P(\overline{A}) = 1 - P(A)\).
• If the Expected Value of a game is zero, we say that it is a 'fair game'