Liam Writes All Of The 3-digit Numbers That Use Only The Digit 1, Only The Digit 2, Or Both Digits 1 And 2. He Writes Each 3-digit Number On A Separate Piece Of Paper. What Is The Probability That The Number Liam Chooses Has At Least One 2?
The Probability of Choosing a Number with at Least One 2
In this problem, we are tasked with finding the probability that a randomly chosen 3-digit number has at least one 2. To approach this problem, we need to first understand the total number of possible 3-digit numbers that Liam can choose from. We will then find the number of 3-digit numbers that have at least one 2 and calculate the probability accordingly.
Total Number of Possible 3-Digit Numbers
A 3-digit number can have any of the digits 0-9 in the hundreds, tens, and ones place. However, since we are dealing with 3-digit numbers, the hundreds place cannot be 0. Therefore, the hundreds place can have 9 possible digits (1-9), and the tens and ones place can have 10 possible digits each (0-9).
Using the fundamental counting principle, we can find the total number of possible 3-digit numbers as follows:
- Number of possible digits for the hundreds place: 9
- Number of possible digits for the tens place: 10
- Number of possible digits for the ones place: 10
Total number of possible 3-digit numbers = 9 Ă— 10 Ă— 10 = 900
Number of 3-Digit Numbers with at Least One 2
To find the number of 3-digit numbers with at least one 2, we can use the complementary counting principle. This principle states that the number of elements in a set A is equal to the number of elements in the universal set U minus the number of elements in the complement of A.
In this case, the universal set U is the set of all possible 3-digit numbers, and the complement of A is the set of all 3-digit numbers with no 2's.
Number of 3-Digit Numbers with No 2's
A 3-digit number with no 2's can have any of the digits 0-1, 3-9 in the hundreds, tens, and ones place. The hundreds place can have 8 possible digits (1, 3-9), and the tens and ones place can have 9 possible digits each (0, 1, 3-9).
Using the fundamental counting principle, we can find the number of 3-digit numbers with no 2's as follows:
- Number of possible digits for the hundreds place: 8
- Number of possible digits for the tens place: 9
- Number of possible digits for the ones place: 9
Number of 3-digit numbers with no 2's = 8 Ă— 9 Ă— 9 = 648
Number of 3-Digit Numbers with at Least One 2
Using the complementary counting principle, we can find the number of 3-digit numbers with at least one 2 as follows:
Number of 3-digit numbers with at least one 2 = Total number of possible 3-digit numbers - Number of 3-digit numbers with no 2's = 900 - 648 = 252
Probability of Choosing a Number with at Least One 2
The probability of choosing a number with at least one 2 is equal to the number of 3-digit numbers with at least one 2 divided by the total number of possible 3-digit numbers.
Probability = Number of 3-digit numbers with at least one 2 / Total number of possible 3-digit numbers = 252 / 900 = 0.28
Therefore, the probability that the number Liam chooses has at least one 2 is 0.28 or 28%.
Q: What is the probability of choosing a number with at least one 2?
A: The probability of choosing a number with at least one 2 is 0.28 or 28%.
Q: How many 3-digit numbers are there in total?
A: There are 900 possible 3-digit numbers, as each digit can have 10 possible values (0-9) and the hundreds place cannot be 0.
Q: How many 3-digit numbers have no 2's?
A: There are 648 3-digit numbers with no 2's, as each digit can have 9 possible values (0, 1, 3-9) and the hundreds place cannot be 0.
Q: How many 3-digit numbers have at least one 2?
A: There are 252 3-digit numbers with at least one 2, which is the difference between the total number of possible 3-digit numbers and the number of 3-digit numbers with no 2's.
Q: What is the difference between the total number of possible 3-digit numbers and the number of 3-digit numbers with no 2's?
A: The difference between the total number of possible 3-digit numbers and the number of 3-digit numbers with no 2's is 252, which is the number of 3-digit numbers with at least one 2.
Q: How was the probability of choosing a number with at least one 2 calculated?
A: The probability of choosing a number with at least one 2 was calculated by dividing the number of 3-digit numbers with at least one 2 (252) by the total number of possible 3-digit numbers (900).
Q: What is the significance of the probability of choosing a number with at least one 2?
A: The probability of choosing a number with at least one 2 is significant because it gives us an idea of the likelihood of a randomly chosen 3-digit number having at least one 2. This can be useful in various applications, such as probability theory, statistics, and data analysis.
Q: Can the probability of choosing a number with at least one 2 be used in real-world scenarios?
A: Yes, the probability of choosing a number with at least one 2 can be used in real-world scenarios, such as:
- Probability theory: The probability of choosing a number with at least one 2 can be used to model real-world events and make predictions.
- Statistics: The probability of choosing a number with at least one 2 can be used to analyze data and make inferences.
- Data analysis: The probability of choosing a number with at least one 2 can be used to identify patterns and trends in data.
Q: Are there any limitations to the probability of choosing a number with at least one 2?
A: Yes, there are limitations to the probability of choosing a number with at least one 2. For example:
- The probability is based on the assumption that each digit is equally likely to be chosen.
- The probability does not take into account any external factors that may affect the choice of a number.
- The probability is only applicable to 3-digit numbers and may not be applicable to other types of numbers.