Questions: A fair bet is a bet where, if repeated, you would end up with the same amount of money as before the bet was accepted.

Transcript text: A fair bet is a bet where, if repeated, you would end up with the same amount of money as before the bet was accepted.

Solution

Solution Steps

To determine what constitutes a fair bet, we need to consider the expected value of the bet. A fair bet is one where the expected value is zero, meaning that over time, you neither gain nor lose money on average. This implies that the probability of winning times the payout should equal the probability of losing times the loss.

Step 1: Define the Variables

Let:

\( P(W) = 0.5 \) (probability of winning)
\( P(L) = 1 - P(W) = 0.5 \) (probability of losing)
\( Payout = 10 \) (amount won if the bet is successful)
\( Loss = 10 \) (amount lost if the bet is unsuccessful)

Step 2: Calculate the Expected Value

The expected value \( E \) of the bet can be calculated using the formula: \[ E = P(W) \cdot Payout - P(L) \cdot Loss \] Substituting the values: \[ E = 0.5 \cdot 10 - 0.5 \cdot 10 = 0 \]

Step 3: Determine if the Bet is Fair

A bet is considered fair if the expected value \( E \) is equal to zero. Since we calculated \( E = 0 \), we conclude that the bet is fair.