Part B There Are Seven Tiles In A Bag, Each With A Letter Written On It. A Tile Is Selected At Random, It Is replaced And Then Another Tile Is Selected. Find The Probability That Both Tiles Have A Different Letter On It. P 0 L y G 0 N

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Introduction

In this problem, we are given a bag containing seven tiles, each with a letter written on it. The letters are P, 0, L, y, G, 0, and N. We are asked to find the probability that both tiles have a different letter on them when selected at random, replaced, and then another tile is selected.

Understanding the Problem

To solve this problem, we need to understand the concept of probability. Probability is a measure of the likelihood of an event occurring. In this case, the event is selecting two tiles with different letters.

Step 1: Counting the Total Number of Possible Outcomes

When we select the first tile, there are 7 possible outcomes, as there are 7 tiles in the bag. Since the tile is replaced before selecting the second tile, the number of possible outcomes remains the same for the second selection.

Step 2: Counting the Number of Favorable Outcomes

To find the number of favorable outcomes, we need to count the number of ways we can select two tiles with different letters. We can do this by considering the following cases:

  • Case 1: The first tile has the letter P, and the second tile has a different letter (0, L, y, G, 0, or N). There are 6 possible outcomes in this case.
  • Case 2: The first tile has the letter 0, and the second tile has a different letter (P, L, y, G, 0, or N). There are 6 possible outcomes in this case.
  • Case 3: The first tile has the letter L, and the second tile has a different letter (P, 0, y, G, 0, or N). There are 6 possible outcomes in this case.
  • Case 4: The first tile has the letter y, and the second tile has a different letter (P, 0, L, G, 0, or N). There are 6 possible outcomes in this case.
  • Case 5: The first tile has the letter G, and the second tile has a different letter (P, 0, L, y, 0, or N). There are 6 possible outcomes in this case.
  • Case 6: The first tile has the letter 0, and the second tile has a different letter (P, L, y, G, 0, or N). There are 6 possible outcomes in this case.
  • Case 7: The first tile has the letter N, and the second tile has a different letter (P, 0, L, y, G, 0). There are 6 possible outcomes in this case.

However, we need to be careful not to double-count the cases. For example, if the first tile has the letter P, and the second tile has the letter 0, we have counted this outcome twice (once in Case 1 and once in Case 2). To avoid this, we can use the concept of combinations to count the number of favorable outcomes.

Step 3: Using Combinations to Count the Number of Favorable Outcomes

We can use the combination formula to count the number of favorable outcomes. The combination formula is given by:

C(n, k) = n! / (k!(n-k)!)

where n is the total number of items, k is the number of items to choose, and ! denotes the factorial function.

In this case, we have 7 tiles, and we want to choose 2 tiles with different letters. We can use the combination formula to count the number of favorable outcomes:

C(7, 2) = 7! / (2!(7-2)!) = 7! / (2!5!) = (7 × 6 × 5 × 4 × 3 × 2 × 1) / ((2 × 1)(5 × 4 × 3 × 2 × 1)) = 21

However, this counts the number of ways to choose 2 tiles with different letters, but it does not take into account the fact that the tiles are replaced before selecting the second tile. To account for this, we need to multiply the number of favorable outcomes by 2:

2 × 21 = 42

Step 4: Finding the Probability

Now that we have counted the number of favorable outcomes, we can find the probability of selecting two tiles with different letters. The probability is given by the ratio of the number of favorable outcomes to the total number of possible outcomes:

Probability = Number of favorable outcomes / Total number of possible outcomes = 42 / (7 × 7) = 42 / 49 = 6/7

Conclusion

In this problem, we found the probability of selecting two tiles with different letters from a bag containing 7 tiles, each with a letter written on it. We used the concept of combinations to count the number of favorable outcomes and found that the probability is 6/7.

Discussion

This problem is a classic example of a probability problem that involves counting and combinations. The concept of combinations is a powerful tool for solving problems that involve counting and arranging objects in different ways.

Real-World Applications

The concept of probability is used in many real-world applications, such as:

  • Insurance: Probability is used to calculate the likelihood of an event occurring, such as a car accident or a natural disaster.
  • Finance: Probability is used to calculate the likelihood of a stock price increasing or decreasing.
  • Medicine: Probability is used to calculate the likelihood of a patient responding to a treatment.
  • Engineering: Probability is used to calculate the likelihood of a system failing or functioning properly.

Future Research Directions

There are many future research directions in the field of probability, including:

  • Developing new methods for calculating probability
  • Applying probability to new fields, such as machine learning and data science
  • Investigating the relationship between probability and other mathematical concepts, such as statistics and combinatorics.

References

  • [1] "Probability and Statistics" by James E. Gentle
  • [2] "Combinatorics: Topics, Techniques, Algorithms" by Peter J. Cameron
  • [3] "Probability: Theory and Examples" by Rick Durrett