Question

The odds in favour of an event \(A\) are 6 to 4 . The probability of success of \(A\) is (a) \(4 / 5\) (b) \(5 / 9\) (c) \(4 / 9 .\)

Short answer

None of the options match the calculated probability \(\frac{3}{5}\).

Step-by-step solution

Understand Odds

The problem states that the odds in favor of event \(A\) are 6 to 4. This means for every 6 successful outcomes, there are 4 failures. Here, successful outcomes are those where event \(A\) happens and failures are when it doesn't.

Express Odds in Terms of Probability

In terms of probability, odds are expressed as the ratio of success to total outcomes. Since there are 6 successful outcomes and 4 failures, the total number of outcomes is \(6 + 4 = 10\). Thus, the probability \(P(A)\) is the number of successful outcomes divided by the total number of outcomes.

Calculate Probability

Using the formula for probability, \(P(A) = \frac{\text{number of successful outcomes}}{\text{total number of outcomes}} = \frac{6}{10}\). Simplifying this fraction gives \(P(A) = \frac{3}{5}\).

Select the Correct Answer

Compare \(\frac{3}{5}\) with the given options: (a) \(\frac{4}{5}\), (b) \(\frac{5}{9}\), and (c) \(\frac{4}{9}\). None of the options match \(\frac{3}{5}\). Therefore, it seems there was a mistake, so revisit the calculation or clarify with the teacher.

Key concepts

Odds Ratio

Odds ratio might sound complicated at first, but it’s really just a way of comparing successful outcomes to failure outcomes. In this exercise, the odds in favor of an event happening were given as 6 to 4. This means for every 6 times the event occurs successfully, it fails 4 times.

• Odds Ratio Definition: It is the ratio of the probability that an event will occur to the probability that it will not occur.
• Calculation: In a situation where the odds are 6 to 4, the odds ratio is 6/4.

Understanding this concept helps in converting odds to probabilities, which is often more intuitive in assessing chances.

Successful Outcomes

Successful outcomes are the events you expect to occur. In probability, these are the outcomes that count towards the event happening. With odds of 6 to 4, we know that there are 6 successful outcomes. If we visualize it:

• Given: 6 successful outcomes.
• Role: For probability calculations, this number goes in the numerator of any probability fraction.

Each successful outcome essentially represents the event completing as desired. Knowing this simplifies the process of determining probabilities based on odds.

Failure Outcomes

Failure outcomes are those that don't result in the event occurring. When odds are given, these are the counterpart to successful outcomes. Here, for every 6 successful outcomes, there are 4 failure outcomes, indicating the event didn't occur.

• Given: 4 failure outcomes.
• Use: These failures are crucial as converting odds to probability requires both successful and failure outcomes.

Failure outcomes help us understand what happens when the event doesn't occur. They balance the equation of the odds ratio by completing the total possible outcomes.

Probability Calculation

Probability calculation is the main goal when determining odds for certain events. Once you have the number of successful and failure outcomes from your odds ratio, calculating the probability is straightforward.
Start by finding the total number of outcomes, which is the sum of both successful and failure outcomes: \[\text{Total Outcomes} = 6 + 4 = 10\]Next step is to calculate the probability of success using this formula:\[P(A) = \frac{\text{number of successful outcomes}}{\text{total number of outcomes}}\]Plug in the numbers:\[P(A) = \frac{6}{10} = \frac{3}{5}\]Despite the correct calculation of \( \frac{3}{5} \), none of the given options match. This indicates a potential error in the options list, not the calculation itself. Probability gives us a clearer picture of how likely an event will occur.