what is meant by favourable outcomes and total outcomes (math probablity)
The Correct Answer and Explanation is:
In probability, the terms favorable outcomes and total outcomes are used to describe the possible outcomes of a random experiment.
- Favorable Outcomes:
- These refer to the specific outcomes that align with the event we are interested in. In other words, favorable outcomes are the outcomes that result in the event we want to occur.
- For example, if you are rolling a fair six-sided die and want to know the probability of rolling a 4, then the favorable outcome is just rolling a 4. If you want to find the probability of rolling an even number, the favorable outcomes would be rolling a 2, 4, or 6.
- In short, favorable outcomes are the “desired” results from a probability experiment.
- Total Outcomes:
- This refers to all possible outcomes that can occur in a given experiment. It includes every possible result that could happen, regardless of whether or not they are favorable to the event you’re interested in.
- Continuing with the die example, the total outcomes are all the possible results from rolling the die, which are 1, 2, 3, 4, 5, and 6. So, the total number of outcomes here is 6.
- In short, total outcomes represent the complete set of possibilities for an experiment.
Example:
If we roll a standard six-sided die:
- The total outcomes are {1, 2, 3, 4, 5, 6}, which gives a total of 6 outcomes.
- If we are interested in rolling an even number, the favorable outcomes are {2, 4, 6}, which gives 3 favorable outcomes.
Probability Calculation:
The probability of an event is calculated as the ratio of favorable outcomes to total outcomes: P(Event)=Number of Favorable OutcomesNumber of Total OutcomesP(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Number of Total Outcomes}}P(Event)=Number of Total OutcomesNumber of Favorable Outcomes
For the example above, the probability of rolling an even number is: P(even)=36=12P(\text{even}) = \frac{3}{6} = \frac{1}{2}P(even)=63=21
In conclusion, favorable outcomes are the ones we are hoping to achieve, and total outcomes represent all possible results of the experiment.
