Probability » Combinations with and without replacing  

With some exercises you have a combination between with and without replacing. Often you use the vase model to calculate the probability and then you use this probability in the product rule.

In a vase are 5 red, 4 white and 3 blue marbles.
John takes 3 marbles in one grab without looking and afterwards replaces these.
He repeats this another 5 times.
Calculate the probability that he had exactly two grabs without a red marble.

The answer can be found by doing the following two steps:
First calculate the probability of grabbing no red marble using the vase model.
bracket open7
bracket closebracket open5
bracket close

bracket open12
bracket close
 ≈ 0.159

After that you calculate the probability of having exactly 2 times not a red marble after 6 grabs. The probability of getting a red marble is 1 – 0.159 = 0.841.
 bracket open6
bracket close(0.159)2(0.841)4 ≈ 0.19