Probability of Drawing Balls of the Same Color from Two Bags

Today, let's delve into the problem of probability by considering two bags of balls. Each bag contains a mix of blue and green balls, and we want to find the probability that balls drawn from each bag are of the same color. This problem is a classic example of conditional probability and the multiplication rule of probability.

Understanding the Problem

We have two bags. The first bag contains 4 blue and 5 green balls, while the second bag has 3 blue and 7 green balls. We need to find the probability that drawing one ball from each bag results in both balls being blue or both being green.

Step 1: Calculate the Total Number of Balls in Each Bag

Let's start by calculating the total number of balls in each bag:

Bag 1: 4 blue balls 5 green balls 9 balls Bag 2: 3 blue balls 7 green balls 10 balls

Step 2: Calculate the Probabilities for Each Scenario

Next, we calculate the probability for each possible scenario where the balls drawn are of the same color.

Both Balls are Blue

We need to find the probability of drawing a blue ball from each bag:

Probability of drawing a blue ball from Bag 1: PBlue from Bag 1 4/9 Probability of drawing a blue ball from Bag 2: PBlue from Bag 2 3/10

The combined probability for both being blue is:

PBoth Blue PBlue from Bag 1 × PBlue from Bag 2 4/9 × 3/10 12/90 2/15

Both Balls are Green

Now, we calculate the probability of drawing green balls from each bag:

Probability of drawing a green ball from Bag 1: PGreen from Bag 1 5/9 Probability of drawing a green ball from Bag 2: PGreen from Bag 2 7/10

The combined probability for both being green is:

PBoth Green PGreen from Bag 1 × PGreen from Bag 2 5/9 × 7/10 35/90 7/18

Step 3: Add the Probabilities of Both Scenarios

To find the total probability that the two balls drawn are of the same color, we add the probabilities of both scenarios:

PSame Color PBoth Blue PBoth Green 2/15 7/18

Step 4: Find a Common Denominator and Compute

The least common multiple of 15 and 18 is 90. We convert the fractions to have a common denominator of 90:

2/15 2 × 6 / 15 × 6 12/90 7/18 7 × 5 / 18 × 5 35/90

Adding these fractions gives us:

PSame Color 12/90 35/90 47/90

Final Answer

The probability that the two balls drawn are of the same color is:

boxed{47/90}

Thus, the probability of drawing balls of the same color from the two bags is 47/90.