Calculating Probabilities: Two-Fruit Selection from a Mixed Basket

Probability calculations are an essential part of understanding and predicting outcomes in various scenarios, from everyday life to scientific research. In this article, we delve into the calculation of probabilities involving the selection of two fruits from a basket containing a mix of apples and oranges. We will use the principle of inclusion-exclusion to find the probability that either both fruits are oranges or both are non-defective.

Setting Up the Scenario

We start with a basket containing 10 apples and 20 oranges. Among these, 3 apples and 5 oranges are defective. Our task is to determine the probability that when two fruits are chosen at random, either both are oranges or both are non-defective.

Step 1: Total Fruits

Total fruits in the basket: 10 apples 20 oranges 30 fruits

Step 2: Total Ways to Choose 2 Fruits

The total number of ways to choose 2 fruits from 30 is calculated using the combination formula:

( binom{n}{r} frac{n!}{r!(n-r)!} )

Using this formula, the total ways to choose 2 fruits from 30 is:

( binom{30}{2} frac{30 times 29}{2} 435 )

Step 3: Probability of Both Fruits Being Oranges

Total oranges in the basket: 20 oranges Ways to choose 2 oranges from 20:

( binom{20}{2} frac{20 times 19}{2} 190 )

Thus, the probability that both fruits are oranges is:

( P_{both,oranges} frac{190}{435} )

Step 4: Probability of Both Fruits Being Non-Defective

Non-defective apples: 10 apples - 3 defective apples 7 non-defective apples Non-defective oranges: 20 oranges - 5 defective oranges 15 non-defective oranges Total non-defective fruits in the basket: 7 apples 15 oranges 22 non-defective fruits Ways to choose 2 non-defective fruits from 22:

( binom{22}{2} frac{22 times 21}{2} 231 )

Thus, the probability that both fruits are non-defective is:

( P_{both,non-defective} frac{231}{435} )

Step 5: Probability of Both Being Oranges and Non-Defective

Since the event of both fruits being oranges is independent of whether they are defective, we do not need to consider any overlap. Therefore, we can simply add the probabilities to find the final answer.

Step 6: Total Probability

Using the principle of inclusion-exclusion, the total probability that either both fruits are oranges or both are non-defective is:

( P_{both,oranges,or,both,non-defective} P_{both,oranges} P_{both,non-defective} - P_{both,oranges,and,non-defective} )

Substituting the values we calculated:

( P_{both,oranges,or,both,non-defective} frac{190}{435} frac{231}{435} - 0 frac{421}{435} )

Final Answer: The probability that either both fruits are oranges or both are non-defective is:

( frac{421}{435} )

Conclusion

Probability is not just a theoretical concept; it has practical applications in various fields, from statistics to engineering. Understanding how to calculate probabilities accurately is crucial for making informed decisions and predicting outcomes. By using the principle of inclusion-exclusion, we can solve complex probability problems efficiently.

Key Points to Remember

The principle of inclusion and exclusion is a powerful tool for solving probability problems where events are not mutually exclusive. The combination formula is essential for calculating the number of ways to choose items from a set. Understanding the independence of events is crucial for simplifying probability calculations.