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A fair die with its faces numbered from 1 to 6 will be rolled. Which of the following is the best interpretation of the probability that the number landing face up will be less than 3?
The best interpretation of the probability that the number landing face up will be less than 3 is 1/3
The probability that the number landing face up will be less than 3 can be interpreted as the likelihood of rolling a number that is either 1 or 2.
To find this probability, we first determine the total number of possible outcomes when rolling a fair six-sided die, which is 6 (since there are 6 possible numbers: 1, 2, 3, 4, 5, and 6).
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Then, we determine the number of favorable outcomes, which are the outcomes where the number landing face up is less than 3, i.e., either 1 or 2. There are 2 favorable outcomes.
So, the probability that the number landing face up will be less than 3 is the ratio of favorable outcomes to total outcomes:
P(number less than 3) = Number of favorable outcomes / Total number of outcomes = 2 / 6 = 1/3
Therefore, the best interpretation of the probability that the number landing face up will be less than 3 is 1/3, meaning there is a 1 in 3 chance of rolling a number less than 3 on a fair six-sided die.
Combinations and Permutations
Combinations and permutations are fundamental concepts in combinatorics, a branch of mathematics concerned with counting, arranging, and organizing objects. They are used to calculate the number of ways objects can be selected or arranged.
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Permutations: Permutations refer to the arrangements of objects in a specific order. The number of permutations of a set of objects is calculated using the formula:
P(n, r) = n! / (n – r)!
Where:
- n is the total number of objects.
- r is the number of objects selected.
- n! represents the factorial of n, which is the product of all positive integers up to n.
In permutations, the order of selection matters.
Combinations: Combinations refer to selections of objects without considering the order. The number of combinations of a set of objects is calculated using the formula:
C(n, r) = n! / (r! * (n – r)!)
Where:
- n is the total number of objects.
- r is the number of objects selected.
- n! represents the factorial of n, which is the product of all positive integers up to n.
- r! represents the factorial of r.
- (n – r)! represents the factorial of n – r.
In combinations, the order of selection does not matter.
To summarize:
- Permutations deal with arrangements where order matters.
- Combinations deal with selections where order does not matter.
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