A bag contains 10 red marbles, 10 green marbles, 10 yellow marbles and 10 blue marbles. You reach into the bag and grab a marble, then reach into the bag and grab a second marble. The probability that the second marble is the same color as the first marble is ab, where a and b are positive, coprime integers. What is the value of a+b?

Note: You do not place the first marble back into the bag

could be RR, GG, YY, or BB

prob RR = (10/40)(9/39) = 3/52

the others are the same calculation
so prob = 4(3/52) = 9/52

now doing the a and b stuff
a=9 , b=52
a+b = 61

it is the wrong answer

The first one can be any color. But after you draw it there will be only 9 of that color left.

So the odds of drawing the same color again are 9/39.

To calculate the probability that the second marble is the same color as the first marble, we need to determine the total number of possible outcomes (the denominator) and the number of favorable outcomes (the numerator).

Let's break down the problem:

First, we need to determine the color of the first marble. There are a total of 40 marbles in the bag, so the probability of selecting any particular color for the first marble is 10/40 = 1/4.

Once we have chosen a first marble, there are now 39 marbles in the bag. If we want to select a marble of the same color for the second choice, there are 9 remaining marbles of that color. So, the probability of choosing a matching color for the second marble, given the color of the first marble, is 9/39 = 3/13.

Since we randomly select the color of the first marble and there are four possible outcomes (red, green, yellow, or blue), we need to multiply the probability of the first marble being a particular color by the probability of the second marble being the same color for each color.

Probability of selecting a red first marble and then a matching color second marble: (1/4) * (3/13) = 3/52
Probability of selecting a green first marble and then a matching color second marble: (1/4) * (3/13) = 3/52
Probability of selecting a yellow first marble and then a matching color second marble: (1/4) * (3/13) = 3/52
Probability of selecting a blue first marble and then a matching color second marble: (1/4) * (3/13) = 3/52

Next, we add up the probabilities of all the possible outcomes:

Probability of selecting a matching color for the second marble = (3/52) + (3/52) + (3/52) + (3/52) = 12/52 = 3/13

Therefore, a = 3 and b = 13. The value of a + b = 3 + 13 = 16.

So, the probability that the second marble is the same color as the first marble is 3/13, and the value of a + b is 16.