Probability Calculation of Drawing Cards with a Sum of 3 from a Box

Let's delve into the probability of drawing two cards from a box containing five cards numbered 1, 1, 2, 2, and 3, with the goal of determining the probability that the sum of the two drawn cards is 3. This problem provides a wonderful opportunity to explore the principles of probability and combinations.

Identifying Possible Outcomes

First, we need to identify the total number of ways to draw 2 cards from 5. This can be calculated using the combination formula:

Total ways 5C2 5! /2!(5-2)! 5 × 4/2 × 1 10

This means there are 10 possible outcomes when drawing 2 cards from the box.

Identifying Successful Outcomes

To find the combinations of cards that sum to 3, we need to consider the possible pairs. The only pairs that can achieve this are:

(1, 2)

Here, we can select one of the two 1s and one of the two 2s. This gives us:

1 from the first 1 and 2 from the first 2 1 from the first 1 and 2 from the second 2 1 from the second 1 and 2 from the first 2 1 from the second 1 and 2 from the second 2

Thus, there are 4 successful outcomes.

It's important to note that (2, 1) is the same as (1, 2) in terms of achieving a sum of 3, so it does not provide additional combinations. Similarly, there are no combinations involving a 3 since we cannot draw a 0.

Calculating the Probability

The probability that the sum of the two drawn cards is 3 is given by the ratio of the number of successful outcomes to the total number of outcomes:

P Number of successful outcomes/Total outcomes 4/10 2/5

Therefore, the probability that the sum of the two numbers drawn from the box is 3 is 2/5.

Alternative Calculation Method

For a different perspective, let's consider the probability calculation step-by-step:

The chance of selecting one of the 1's on the first draw is 2 in 5. After selecting one of the 1's, the chance of selecting one of the 2's on the next draw is 2 in 4.

By multiplying these probabilities, we find:

(2/5) × (2/4) (4/20) (1/5)

However, we must also consider the reverse order of selection (2, 1), which has the same probability.

Thus, the overall probability is:

(1/5) (1/5) 2/5

Both methods reaffirm that the probability of drawing two cards with a sum of 3 is 2/5.