Question

which of the following statements are true? mark all that apply.
if the expected value of a game is zero, we say that it is a fair game
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) ).
the probability of an event that is certain is 0.
expected value is the actual gain or loss each time an event takes place.
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.
question help: message instructor

Explanation:

• First statement: By definition, a game with an expected value of zero is fair.
• Second statement: The sum of the probabilities of an event occurring and not occurring is 1. So, \(P(\overline{A}) = 1 - P(A)\).
• Third statement: The probability of a certain event is 1, not 0.
• Fourth statement: Expected value is an average, not the actual gain/loss each time.
• Fifth statement: Expected value is the average gain or loss over many repetitions.
• Sixth statement: If odds for an event are \(a:b\), odds against are \(b:a\). Here, 3 to 5 for means 5 to 3 against.
• Seventh statement: Probability of an impossible event is 0, not - 1.

Answer:

• If the Expected Value of a game is zero, we say that it is a 'fair game'
• 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 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