Points to Remember:
- Basic probability principles: The probability of an event is the ratio of favorable outcomes to the total number of possible outcomes.
- Sample space: The set of all possible outcomes of an experiment.
- Complementary events: Events that together encompass the entire sample space.
Introduction:
This question pertains to probability, a branch of mathematics dealing with the likelihood of events occurring. When three coins are tossed, we are interested in determining the probability of a specific outcome: getting neither three heads (HHH) nor three tails (TTT). This requires understanding the sample space of all possible outcomes and applying basic probability calculations.
Body:
1. Defining the Sample Space:
When three coins are tossed, the sample space consists of all possible combinations of heads (H) and tails (T). This can be represented as:
{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
There are a total of 2³ = 8 possible outcomes.
2. Identifying Favorable Outcomes:
We are interested in the outcomes that are neither three heads nor three tails. This means we exclude HHH and TTT from our sample space. The remaining outcomes are:
{HHT, HTH, HTT, THH, THT, TTH}
There are 6 favorable outcomes.
3. Calculating the Probability:
The probability of getting neither three heads nor three tails is the ratio of favorable outcomes to the total number of possible outcomes:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes) = 6/8 = 3/4 = 0.75
Therefore, the probability is 0.75 or 75%.
Conclusion:
In conclusion, the probability of getting neither three heads nor three tails when tossing three coins is 3/4 or 75%. This is calculated by identifying the total number of possible outcomes (8) and the number of outcomes that meet the specified condition (6), and then finding the ratio. This simple example demonstrates the fundamental principles of probability, which have wide-ranging applications in various fields, from risk assessment to scientific research. A deeper understanding of probability helps in making informed decisions based on the likelihood of different events. Further exploration into conditional probability and other advanced concepts can enhance our ability to analyze complex scenarios and make more accurate predictions.
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