Let’s understand another example on a new page. Assume you have three red marbles and three blue marbles in a box. Your task is to pull out two marbles out of these six marbles, one after the other. This example is interesting, but I need you to focus really really hard. So there will be two cases here! First case, you pick the first marble out and then put it back and then take the second marble out. You take one out, put it back in and then pick the second marble. Let’s talk about this case first. Say we want to find the probability of getting a red marble out in your first chance. Let’s call it P of R1. There are three red marbles and six marbles in all. So the probability of getting a red marble is ‘three by six’. The numerator tells us that we can pick a red marble in three ways and the denominator tells us the total number of marbles. So you picked one red marble up and placed it back in. Now what is the probability of picking a red marble in your second chance? We call it P of R2. Since we place the ball back in there are three red marbles and six total marbles. So the probability still remains ‘three by six’, it’s unchanged. So these two events are called independent events. The probability does not change if the marble is replaced. It’s because we have the same set of marbles we had originally. Now let’s look at the second case. You pick one marble out and then pick the second one out without putting it back in. Now it gets even more interesting. Focus hard. In this case we pick one marble, we do not put it back in and then we pick a second marble. Now let’s find the probability of picking a red marble in your first chance. It will be ‘three by six’. You can pick a red marble in three ways out of the six possible marbles. So assume the red marble is taken out. But remember, this time it’s not put back in the box. It remains outside. Now what is the probability of picking a red marble? There are two red marbles and five in total. So the probability of picking a red marble is ‘two by five’. Did you observe what happened? We can see that the probability has changed if the marble is not replaced. So in this case, the events are not independent. Now forget the colour of the marbles. You should know that if a marble is put back in, then the events will be independent as the sample space does not change for the second case. But if the marble is not put back in, then the sample space changes for the second event which is why the events are not independent. The probability of second event actually depends on what marble you have picked first. We will understand what mutually exclusive means in our next video.