Points to Remember:
- Understanding of probability calculations.
- Application of combination formula.
- Accurate calculation of the total number of ways to choose 5 balls.
- Accurate calculation of the number of ways to choose 5 white balls.
Introduction:
This question requires a factual and analytical approach to determine the probability of a specific eventâdrawing five white balls from a basket containing balls of different colors. Probability, in its simplest form, is the ratio of favorable outcomes to the total number of possible outcomes. In this case, the favorable outcome is drawing five white balls, and the total number of possible outcomes is the number of ways to choose any five balls from the basket. We will use the principles of combinations (choosing items without regard to order) to solve this problem.
Body:
1. Calculating the Total Number of Possible Outcomes:
The basket contains a total of 3 + 7 + 2 = 12 balls. We need to find the number of ways to choose 5 balls out of 12. This is given by the combination formula:
nCr = n! / (r! * (n-r)!)
where n is the total number of items (12 balls), and r is the number of items we choose (5 balls).
Therefore, the total number of ways to choose 5 balls from 12 is:
12C5 = 12! / (5! * 7!) = 792
2. Calculating the Number of Favorable Outcomes:
The favorable outcome is choosing 5 white balls. There are 7 white balls in the basket. The number of ways to choose 5 white balls from 7 is:
7C5 = 7! / (5! * 2!) = 21
3. Calculating the Probability:
The probability of drawing 5 white balls is the ratio of the number of favorable outcomes to the total number of possible outcomes:
Probability = (Number of ways to choose 5 white balls) / (Total number of ways to choose 5 balls)
Probability = 21 / 792 = 7/264
Conclusion:
The probability of drawing 5 white balls from a basket containing 3 red, 7 white, and 2 blue balls is 21/792, which simplifies to 7/264. This represents a relatively low probability, indicating that this event is not very likely to occur. This calculation demonstrates a fundamental application of probability theory in a simple scenario. Further applications of probability theory are crucial in various fields, including risk assessment, statistical analysis, and decision-making processes. A deeper understanding of probability enhances our ability to make informed choices and predictions based on available data. The focus should always be on accurate data collection and rigorous application of mathematical principles to ensure reliable results.
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