A jar contains 5 red marbles, numbered 1 to 5, and 12 blue marbles numbered 1 to 12.

a) A marble is chosen at random. If you're told the marble is red, what is the probability that it has the number 3 on it?

b) The first marble is replaced, and another marble is chosen at random. If you're told the marble has the number 1 on it, what is the probability the marble is red?

a) Well, if you're told the marble is red, that already narrows down the options. There are a total of 5 red marbles, but only 1 of them has the number 3 on it. So the probability of it being red and having the number 3 on it is 1/5. And by the way, that's also how many licks it takes to get to the center of a Tootsie Pop. Just in case you were wondering.

b) Ah, the old switcheroo! If the first marble is replaced and another one is chosen, we still have the same number of marbles. There are still 5 red marbles and 12 blue marbles. However, since we know this time that the marble has the number 1 on it, we can narrow it down. Only the blue marbles have the number 1, so we can safely say that the marble is blue. So the probability of it being red is 0. Sorry, no red for you this time! But hey, blue is the color of the sky and the ocean, so it's not all bad!

a) To find the probability that a randomly chosen red marble has the number 3 on it, we need to find the number of favorable outcomes and the number of total outcomes.

The favorable outcome is choosing a red marble with the number 3 on it. There is only 1 red marble with the number 3.

The total outcome is choosing any red marble. There are a total of 5 red marbles.

Therefore, the probability that a randomly chosen red marble has the number 3 on it is 1/5.

b) To find the probability that a randomly chosen marble with the number 1 on it is red, we need to find the number of favorable outcomes and the number of total outcomes.

The favorable outcome is choosing a red marble with the number 1 on it. There is only 1 red marble with the number 1.

The total outcome is choosing any marble with the number 1 on it. There is 1 red marble with the number 1 and 1 blue marble with the number 1, so there are 2 marbles with the number 1.

Therefore, the probability that a randomly chosen marble with the number 1 on it is red is 1/2.

To find the probability in both cases, we need to count the number of favorable outcomes and divide it by the total number of possible outcomes.

a) For part a), we want to find the probability of choosing a red marble with the number 3 on it, given that it is red.

There are 5 red marbles in total, and only one of them has the number 3 on it. So the number of favorable outcomes is 1.

Since we are told that the marble is red, the total number of possible outcomes is the total number of red marbles, which is 5.

Therefore, the probability that the red marble has the number 3 on it is 1/5.

b) For part b), we want to find the probability of choosing a red marble, given that the marble has the number 1 on it.

There are 5 red marbles and 12 blue marbles in total. The total number of possible outcomes is the sum of the red and blue marbles, which is 5 + 12 = 17.

Since we are told that the marble has the number 1 on it, we only need to consider the red marbles with the number 1 on it, which is 1.

Therefore, the probability that the marble is red, given that it has the number 1 on it, is 1/17.

a) one of the five red marbles has the number 3 on it, so the probability is 1/5

b) two marbles have the number 1 on them, one red and one blue
... so the probability is 1/2