When determining the probability of an event, it is essential to understand that probability quantifies the likelihood of the occurrence of an event. This likelihood is measured by a number that falls between 0 and 1, inclusive. Let's delve into this in a detailed manner:
1. Definition of Probability:
- Probability ([tex]\( P \)[/tex]) of an event is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes. It is a measure of how likely an event is to occur.
- Mathematically, if an event [tex]\( A \)[/tex] has [tex]\( n \)[/tex] favorable outcomes in a sample space of [tex]\( N \)[/tex] possible outcomes, then the probability of [tex]\( A \)[/tex] is given by:
[tex]\[
P(A) = \frac{\text{Number of favorable outcomes for } A}{\text{Total number of possible outcomes}} = \frac{n}{N}
\][/tex]
2. Probability Values:
- 0 (Zero Probability): A probability of 0 indicates that the event is impossible. There are no favorable outcomes for this event within the given sample space. For example, the probability of rolling a 7 on a standard six-sided die is 0 because it is impossible.
- 1 (Certain Probability): A probability of 1 indicates that the event is certain to occur. All possible outcomes are favorable for the event. For example, the probability of drawing a card from a deck and getting a card that is either red or black in a standard deck (excluding jokers) is 1 (since all cards are either red or black).
- Between 0 and 1: Any probability value between 0 and 1 indicates that there is a certain degree of likelihood that the event will occur. The closer the probability is to 1, the more likely the event is to occur. Conversely, the closer the probability is to 0, the less likely the event is to occur.
3. Why Between 0 and 1:
- The total number of possible outcomes ([tex]\( N \)[/tex]) represents the sample space, and the number of favorable outcomes ([tex]\( n \)[/tex]) can range from 0 to [tex]\( N \)[/tex].
- Consequently, the ratio [tex]\( \frac{n}{N} \)[/tex], which represents the probability, must logically fall between 0 and 1, inclusive.
- Probability as a ratio: Since [tex]\( n \)[/tex] (favorable outcomes) can never exceed [tex]\( N \)[/tex] (total outcomes), the ratio [tex]\( \frac{n}{N} \)[/tex] is always less than or equal to 1. Similarly, since [tex]\( n \)[/tex] cannot be less than 0 (an event cannot have fewer than 0 favorable outcomes), the probability cannot be less than 0.
In conclusion, the reason why the probability of an event must be between 0 and 1 is rooted in the very definition of probability as a ratio of favorable outcomes to total possible outcomes. This bounded range ensures that probability accurately reflects the extent to which an event is likely or unlikely to occur.
