Probability of Summing to 12, 13, or 14 with One Known Ball Number 7

In a scenario where a box contains 10 balls numbered from 1 to 10, and two balls are drawn with one of the balls showing the number 7, we explore the probability of the sum of the two balls being 12, 13, or 14. This problem requires an understanding of probability and conditional probability.

H1: Understanding the Problem

The box contains 10 balls, each numbered from 1 to 10. Two balls are drawn from the box, and one of the drawn balls shows the number 7. We need to determine the probability that the sum of the numbers on the two balls is either 12, 13, or 14. Let's explore the step-by-step reasoning:

Step-by-Step Reasoning

H3: Total Number of Possible Outcomes

The first step is to understand the total number of possible outcomes for the second ball. Once the first ball (number 7) is drawn, there are 9 remaining balls, each with an equal probability of being the second ball. Therefore, there are 9 possible outcomes for the second draw.

Probability of Sums 12, 13, or 14

H3: The Sum is 12

For the sum to be 12, the second ball must be a 5. The probability that the second ball is a 5, given that the first ball is 7, is 1 out of the 9 remaining balls. This can be expressed as:

[ P(text{Sum is 12}) frac{1}{9} ]

H3: The Sum is 13

For the sum to be 13, the second ball must be a 6. The probability that the second ball is a 6, given that the first ball is 7, is also 1 out of the 9 remaining balls. This can be expressed as:

[ P(text{Sum is 13}) frac{1}{9} ]

H3: The Sum is 14

For the sum to be 14, the second ball must be a 7, but we already know one of the balls is 7, so the second ball cannot be 7. Therefore, the probability is 0.

[ P(text{Sum is 14}) 0 ]

Calculating the Total Probability

The total probability of the sum being 12, 13, or 14 is the sum of the individual probabilities:

[ P(text{Sum is 12}) P(text{Sum is 13}) P(text{Sum is 14}) frac{1}{9} frac{1}{9} 0 frac{2}{9} ]

Conclusion

The final answer is that the probability of the sum of the two balls being 12, 13, or 14 is 2/9. This problem illustrates the principles of conditional probability and combinatorics, providing a valuable exercise in probability theory.

References

For further reading on probability and conditional probability, consider the following resources:

Probability Theory by S. Ross Introduction to Probability by K. Ross and C. Anderson Introductory Statistics by T. Hogg, J. McKean, and W. Tannis

Understanding these concepts is crucial for anyone delving into more advanced statistical and mathematical applications.