The probability of rolling a number less than 5 on a standard six-sided die is 2/3.
Understanding Probability
Probability is a fundamental concept in mathematics that quantifies the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. The formula for calculating probability is:
$$P(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}$$
Calculating the Probability
To determine the probability of rolling a number less than 5 on a six-sided die, we need to identify both the total possible outcomes and the favorable outcomes.
Total Possible Outcomes
When a standard six-sided die is rolled once, the face that lands up can be any integer from 1 to 6. These are:
- 1
- 2
- 3
- 4
- 5
- 6
Therefore, the total number of possible outcomes is 6.
Favorable Outcomes (Numbers Less Than 5)
We are interested in outcomes where the number rolled is less than 5. These numbers from our possible outcomes are:
- 1
- 2
- 3
- 4
Thus, the number of favorable outcomes is 4.
Applying the Probability Formula
Using the formula for probability:
- Number of Favorable Outcomes = 4
- Total Number of Possible Outcomes = 6
$$P(\text{Number < 5}) = \frac{4}{6}$$
This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
$$P(\text{Number < 5}) = \frac{4 \div 2}{6 \div 2} = \frac{2}{3}$$
Summary of Outcomes
For clarity, here's a breakdown of the outcomes:
| Type of Outcome | Outcomes | Count |
|---|---|---|
| All Possible | {1, 2, 3, 4, 5, 6} | 6 |
| Favorable | {1, 2, 3, 4} | 4 |
This calculation demonstrates that out of six equally likely possibilities, four of them meet the condition of being less than 5, resulting in a probability of 2/3.
