Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

If \(P(E)=\frac{5}{7}\), find \(O(E)\) and \(O\left(E^{\prime}\right)\)

Step-by-Step Solution

Answer

\(O(E) = \frac{5}{2}\), \(O(E') = \frac{2}{5}\).

Step 1: Understand Probability and Odds

Probability is a measure of how likely an event is to occur, while odds are a different way to express that likelihood. The odds in favor of an event are calculated as the probability of the event occurring divided by the probability of it not occurring.

Step 2: Calculate Probability of Complementary Event

The probability of the complementary event, denoted as \( P(E') \), can be found using the formula: \( P(E') = 1 - P(E) \). Thus, \( P(E') = 1 - \frac{5}{7} \).

Step 3: Compute Complementary Probability

Calculate \(1 - \frac{5}{7}\):\[P(E') = 1 - \frac{5}{7} = \frac{7}{7} - \frac{5}{7} = \frac{2}{7}\]

Step 4: Calculate Odds for Event

The odds in favor of an event \(E\) is given by the ratio \( \frac{P(E)}{P(E')} \). Using the probabilities found:\[O(E) = \frac{ \frac{5}{7} }{ \frac{2}{7} } = \frac{5}{2}\]

Step 5: Calculate Odds Against Complementary Event

The odds against the complementary event \(E'\) is the same calculation as the odds for event \(E\):\[O(E') = \frac{P(E')}{P(E)} = \frac{ \frac{2}{7} }{ \frac{5}{7} } = \frac{2}{5}\]

Key Concepts

Probability of an Event

Probability helps us understand the likelihood that a specific event will occur. It's a number between 0 and 1, where 0 means the event will not happen, and 1 means it certainly will. For example, if the probability of an event is \( P(E) = \frac{5}{7} \), it means there are 5 chances out of 7 for the event to happen. To think about it in everyday terms, imagine you have 7 cards—5 of them are winning cards—and you pick one at random. The probability that you select a winning card is \( \frac{5}{7} \). This is quite a handy way to measure how likely or unlikely events are in our day-to-day lives.

Complementary Event

The complementary event of any event \(E\) is what occurs if \(E\) does not happen. The probability of this complementary event is denoted by \( P(E') \) or sometimes \( \overline{E} \). You can calculate it by subtracting the probability of the event from 1: \( P(E') = 1 - P(E) \). In our example, \( P(E') \) would be calculated as \( 1 - \frac{5}{7} = \frac{2}{7} \). This tells us there's a \( \frac{2}{7} \) chance that the event \(E\) will not occur. Complementary probabilities are always in complete contrast to the original event probabilities and they sum up to 1, helping us see both sides of the chance coin.

Odds in Favor

Odds provide another perspective on chance. They compare the probability of an event occurring to the probability of it not occurring. The odds in favor of an event, \(E\), can be determined using the ratio of its probability to that of its complementary event: \( O(E) = \frac{P(E)}{P(E')} \). In the given example, the odds in favor are \( \frac{ \frac{5}{7} }{ \frac{2}{7} } = \frac{5}{2} \). This tells us that for every 2 times the event does not happen, it is likely to happen 5 times. Simplifying these odds often helps in understanding and predicting outcomes, especially in scenarios like games and betting where understanding odds clearly can make a big difference.

Odds Against

The odds against an event is just the flip of the odds in favor. It represents how often the event will not occur compared to how often it will. For any event \(E\), the odds against are calculated using the ratio \( O(E') = \frac{P(E')}{P(E)} \). From our example, this would be computed as \( \frac{ \frac{2}{7} }{ \frac{5}{7} } = \frac{2}{5} \). This means for every 5 times the event happens, it doesn't happen 2 times. Easily understanding odds against helps in knowing the unfavorable scenarios, and provides an alternate viewpoint on the balance between likelihoods.