Probability of Alice Picking an Even Number and Bob Picking an Odd Number

In this article, we’ll explore how to calculate the probability of Alice picking an even number and Bob picking an odd number when both are randomly selecting a number from 0 to 9. This problem involves understanding independent events and the basic principles of probability.

Understanding the Problem

The problem can be broken down into simpler steps to find a solution. Let’s first identify the total number of possible outcomes and then the number of favorable outcomes.

Total Number of Outcomes

Each participant can pick any integer from 0 to 9. Since there are 10 choices for Alice and 10 choices for Bob, the total number of possible outcomes is:

10 × 10 100

Favorable Outcomes

We need to identify the favorable outcomes for the event where Alice picks an even number and Bob picks an odd number. Let’s start by listing the even and odd numbers in the range from 0 to 9:

Even numbers for Alice: 0, 2, 4, 6, 8 (5 even numbers) Odd numbers for Bob: 1, 3, 5, 7, 9 (5 odd numbers)

Calculating the Number of Favorable Outcomes

Since the choice of Alice and Bob are independent, the number of favorable outcomes is the product of the number of choices for Alice and Bob:

5 (even choices for Alice) × 5 (odd choices for Bob) 25

Calculating the Probability

To find the probability, we use the formula for the ratio of the number of favorable outcomes to the total number of outcomes:

P frac{25}{100} frac{1}{4}

Thus, the probability that Alice picks an even number and Bob picks an odd number is frac{1}{4}.

Additional Considerations

It’s important to note that the calculated probability assumes both participants can choose any number from 0 to 9, including the possibility of them picking the same number. If Bob cannot pick the number Alice picked, the scenarios change significantly:

Alice still chooses from 5 even numbers (0, 2, 4, 6, 8). Bob, however, only has 9 possible choices if Alice has already chosen a number.

In this case, the probability for Bob choosing an odd number would adjust to 5 out of the remaining 9 numbers. The overall probability would be:

frac{5}{10} × frac{5}{9} frac{25}{90}

Conclusion

The problem of finding the probability of Alice picking an even number and Bob picking an odd number is a classic example of working with independent events. By understanding the total number of outcomes and the favorable outcomes, we can derive the probability through simple arithmetic. This article also highlights the importance of considering the context and constraints when dealing with probability problems.