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Answer the given questions that involve odds. When the horse California Chrome won the 140th Kentucky Derby, a \(\$ 2\) bet on a California Chrome win resulted in a winning ticket worth \(\$ 7\). a. How much net profit was made from a \(\$ 2\) win bet on California Chrome? b. What were the payoff odds against a California Chrome win? c. Based on preliminary wagering before the race, bettors collectively believed that California Chrome had a \(0.228\) probability of winning. Assuming that \(0.228\) was the true probability of a California Chrome victory, what were the actual odds against his winning? d. If the payoff odds were the actual odds found in part (c), what would be the worth of a \(\$ 2\) win ticket after the California Chrome win?

Step by step solution

01

Find the Net Profit

To find the net profit, subtract the initial bet from the winning amount. Here, the initial bet is \(\text{\textdollar} 2\) and the winning ticket is worth \(\text{\textdollar} 7\). So, the net profit is \(\text{\textdollar} 7 - \text{\textdollar} 2 = \text{\textdollar} 5\).

02

Calculate Payoff Odds Against a Win

Payoff odds are calculated based on the net profit per dollar wagered. The net profit is \(\text{\textdollar} 5\) for a \(\text{\textdollar} 2\) bet. So, the payoff odds against a win are \(5:2\).

03

Determine Actual Odds Based on Probability

The actual odds against winning can be calculated using the formula \(\text{Odds} = \frac{1 - p}{p}\), where \(p\) is the probability of winning. Given that \(p = 0.228\), the actual odds are \(\frac{1 - 0.228}{0.228} \approx 3.39:1\).

04

Calculate Worth of Win Ticket Based on Actual Odds

If the payoff odds were the actual odds found in part (c), the worth of the winning ticket can be calculated by multiplying the bet with the sum of actual odds and 1. For a \(\text{\textdollar} 2\) bet with actual odds \(3.39:1\), the worth is \(\text{\textdollar} 2 \times (3.39 + 1) = \text{\textdollar} 8.78\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Net Profit

Net profit is a key concept when calculating winnings from a bet. It tells you how much money you earn after you subtract your initial stake. In our example, the initial bet is \(\text{\textdollar} 2\), and the winning ticket is worth \(\text{\textdollar} 7\).
This means the net profit is calculated as \(\text{\textdollar} 7 - \text{\textdollar} 2 = \text{\textdollar} 5\). So, by betting \(\text{\textdollar} 2\) on California Chrome, you would make a net profit of \(\text{\textdollar} 5\).
This method of calculating net profit is essential to understanding how much you actually gain from a bet.

Payoff Odds

Payoff odds give you an idea of your potential earnings compared to your initial bet. They are expressed in the form of `net profit per initial wager`. From our exercise, we calculated a net profit of \(\text{\textdollar} 5\) for a \(\text{\textdollar} 2\) bet.
So, the payoff odds against a win can be expressed as \(5:2\). It means you earn \(\text{\textdollar} 5\) for every \(\text{\textdollar} 2\) you wager.
Understanding payoff odds is crucial for assessing the potential rewards versus the risks of a bet.

Probability

Probability helps us understand the likelihood of an event happening. In our scenario, it was given that bettors collectively believed California Chrome had a \(0.228\) probability of winning the race.
Probability ranges from 0 to 1, where 0 means an impossible event and 1 means a certain event. A probability of \(0.228\) indicates that there is a \(22.8\%\) chance of California Chrome winning.
The formula to convert this probability into odds is \[\text{Odds} = \frac{1 - p}{p}\]. Plugging in \(0.228\) we get \[\frac{1 - 0.228}{0.228} \approx 3.39:1\]. This tells us the actual odds of California Chrome winning, based on the probability.

Actual Odds

Actual odds are calculated based on the true probability of an event. In this exercise, we determined the actual odds to be approximately \(3.39:1\).
This means for every unit of success, there are \(3.39\) units of failure. Suppose the payoffs are consistent with these actual odds, your betting returns will reflect this calculation.
If the actual odds were as calculated, a winning bet of \(\text{\textdollar} 2\) would result in a ticket worth \(\text{\textdollar} 8.78\), calculated by multiplying the bet with the sum of actual odds and 1: \[\text{\textdollar} 2 \times (3.39 + 1) = \text{\textdollar} 8.78\].
This provides a realistic expectation of betting returns based on true probabilities.