Math Problem Statement
A grocery store manager took inventory of all the fruit to decide what needed to be donated. Of the 31 pieces of fruit deemed "overripe," 4 were pears and 17 were peaches.
If the manager randomly selected 6 of the overripe pieces for the first donation package, what is the probability that 3 of the chosen fruits are pears and 3 are peaches?
Write your answer as a decimal rounded to four decimal places.
Solution
To solve this problem, we need to calculate the probability that 3 pears and 3 peaches are selected from the overripe fruits. Here's the breakdown:
Step 1: Total Fruits and Selections
- Total number of overripe fruits =
- Number of pears =
- Number of peaches =
- Total number of fruits to select =
- We want to select 3 pears and 3 peaches.
Step 2: Use of Combinations
We will use the combination formula, denoted as , which is:
Where is the total number of items, and is the number of items to choose.
Step 3: Calculate the Number of Ways to Select the Fruits
-
Number of ways to choose 3 pears from 4 pears:
-
Number of ways to choose 3 peaches from 17 peaches:
-
Total number of ways to select 6 fruits from 31 fruits:
Step 4: Calculate the Probability
The probability is the ratio of the number of favorable outcomes to the total possible outcomes:
Thus, the probability that exactly 3 pears and 3 peaches are selected is approximately .
Would you like more details or have any questions?
Here are 5 related questions to explore:
- What if 4 peaches and 2 pears were selected instead?
- How would the probability change if there were 6 pears and 17 peaches?
- What is the probability of selecting 2 pears and 4 peaches?
- How many different ways can 4 peaches be selected from the group of 17 peaches?
- What if we selected 7 fruits instead of 6?
Tip: In probability, always ensure that you count the favorable outcomes and the total outcomes correctly using combinations when dealing with selections!
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Math Problem Analysis
Mathematical Concepts
Probability
Combinatorics
Formulas
Combination formula: C(n, k) = n! / (k!(n - k)!)
Probability formula: Probability = (favorable outcomes) / (total outcomes)
Theorems
Basic Counting Principle
Combination Theorem
Suitable Grade Level
Grades 10-12
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