91Ó°ÊÓ

To figure out how much money you actually made from a bet, you need to calculate the net profit. In the given exercise, a \$2\ bet on California Chrome resulted in a winning ticket worth \7 dollars. Net profit is simply the winnings minus the amount of the initial bet.

Here is the formula you use:
Net Profit \[ = \text{Winning ticket value} - \text{Initial bet}\]

In this case, it would look like: \[ 7 - 2 = \$5 \]
So, the net profit made is \5 dollars.

This calculation is important because it tells you how much you gained from your investment, excluding the original amount spent.

Payoff odds give you an idea of how much money you can make for each dollar you bet. It's calculated using the net profit and the initial bet amount. Here’s the formula:

\[ \text{Payoff Odds} = \frac{\text{Net Profit}}{\text{Initial Bet}} \]
For the California Chrome winning bet, the net profit was \5 dollars and the initial bet was \2 dollars:

\[ \text{Payoff Odds} = \frac{5}{2} = 2.5:1 \]
What this means is that for every dollar you bet, you make \2.5 dollars in profit. Payoff odds help gamblers understand the potential return on their bets.

Probability is a way to quantify the chance of an event happening. It's expressed as a number between 0 and 1. A probability of 0 means the event will not happen, and a probability of 1 means it will definitely happen.

For California Chrome, the probability of winning was given as 0.228. To convert this into odds against winning, you use the following formula:

\[ \text{Odds against winning} = \frac{\text{Probability of losing}}{\text{Probability of winning}} \]
Since the probability of losing is the complement of the probability of winning (1 - probability of winning), it becomes:
\[ \frac{1 - 0.228}{0.228} = \frac{0.772}{0.228} \approx 3.39:1 \]
This tells us that the odds against California Chrome winning were approximately 3.39 to 1.

In betting, understanding the actual odds against winning is crucial for making informed decisions. Using the probability of winning, you can convert this information into actual odds. As calculated earlier, the actual odds against California Chrome winning were 3.39:1.

To determine what the value of a winning ticket would be if these odds were used, you can use the actual odds to calculate the final payout amount. Here's the formula:

\[ \text{Worth of a win ticket} = \text{Initial Bet} + (\text{Initial Bet} \times \text{Actual Odds}) \]
Plugging the values into the formula, we get:
\[ 2 + (2 \times 3.39) = 2 + 6.78 = \$8.78 \]
This means if someone had bet \2 dollars on California Chrome with the actual odds, their ticket would be worth \8.78 dollars. This understanding helps in comparing the potential returns in different betting scenarios.