Exploring Probability: Selection of Letters from LIME
In this article, we will delve into the principles of probability by examining the selection of two letters from the word LIME. We will consider three different sampling methods: selection without replacement, selection with replacement, and sequential selection. We will then calculate and discuss the probabilities in each scenario.
Case 1: Selection without Replacement
Total Outcomes
Let's start by determining the total number of ways to select two letters from the four letters in LIME. Since the order of selection does not matter, we need to use the combination formula.
Given that we have 4 letters and we want to choose 2, the number of ways to do this is calculated using the combination formula:
nc2 4! / (2! * (4-2)!) (4 * 3) / (2 * 1) 6
Favorable Outcomes
The favorable outcomes are the cases where we select the letters L and M. Considering the word LIME, there is only one way to choose L and M together without replacement.
Calculating the Probability
The probability is given by the formula:
P(L and M) Number of favorable outcomes / Total number of outcomes 1 / 6
Therefore, the probability that the two letters chosen are L and M is .
Case 2: Sequential Selection Without Replacement
Total Outcomes
This time, we will consider the order in which the letters are selected, meaning the sample space is ordered. Here, we have 4 choices for the first letter and 3 remaining choices for the second letter.
Thus, the total number of ways to select two letters in sequence is:
Total Outcomes 4 * 3 12
Favorable Outcomes
Considering the sequential selection, the letters L and M can be chosen in two ways: L followed by M and M followed by L. Therefore, there are 2 favorable outcomes.
Calculating the Probability
The probability is given by the formula:
P(L and M) Number of favorable outcomes / Total number of outcomes 2 / 12 1 / 6
Hence, the probability that the two letters chosen in sequence are L and M is .
Case 3: Selection with Replacement
Total Outcomes
For this scenario, we assume that the letters are drawn one by one with replacement. This means that after selecting a letter, it is put back into the pool before the next selection.
The total number of ways to select two letters with replacement is:
Total Outcomes 4 * 4 16
Favorable Outcomes
Since we are selecting with replacement, the letters L and M can be chosen in 2 ways: LM and ML. Therefore, there are 2 favorable outcomes.
Calculating the Probability
The probability is given by the formula:
P(L and M) Number of favorable outcomes / Total number of outcomes 2 / 16 1 / 8
Hence, the probability that the two letters chosen with replacement are L and M is .
Conclusion
Through the analysis of these three different sampling methods, we have observed how the probability of selecting the letters L and M varies based on whether the selection is with or without replacement and whether the order of selection is considered or not. The key takeaway is that probability calculations can vary significantly based on these factors.