 Hello and welcome to the session. In this session we will discuss the question which says that Martina placed 7 pencils and 5 pens inside her bag. She selected one writing tool without looking and then replaced it and then selected a second writing tool. What is the probability that each writing tool selected is a pen and what will be the probability if she does not replace the first pen. Now before starting the solution of this question we should know some results. First is if A and B are two independent events then probability of the event A and B is equal to probability of the event A into probability of the event B and if A and B are dependent events then probability of event A and B is equal to probability of the event A into probability of the event B following the event A. Now these results will work out as a key idea for solving out the given question. Now let us start with the solution of the given question. Now in the question it is given that Martina has 7 pencils and 5 pens inside her bag. It means in total she has 7 plus 5 is equal to 12 writing tools and it is also given that she selected one writing tool without looking and then replaced it and then selected a second writing tool and we have to find that what is the probability that each writing tool selected is a pen. Now let the event A be drawing a pen and event B is again drawing a pen. Now here she selects one writing tool and puts it back into the bag and then selects the other writing tool. It means here the event A and event B are independent events. Now here the number of pens are 5 so in the first draw number of pens is equal to 5 and total writing tools are 12 which means total number of outcomes is equal to 12 so probability of drawing a pen in a first draw that is probability of the event A is equal to number of favorable outcomes which is 5 upon total number of outcomes which is 12. Now since A and B are independent events so number of pens and total number of outcomes in second draw will remain same. Now in the second draw number of pens will be 5 and total number of outcomes will be 12 so probability of drawing a pen in a second draw that is probability of the event B is equal to 5 upon 12. Now using the first result which is given in the key idea we can find the probability of two independent events so probability of event A and B is equal to probability of event A which is 5 upon 12 into probability of event B which is again 5 upon 12 this is equal to 25 upon 144. So probability that each writing tool selected is a pen is equal to 25 upon 144. Now it is also asked in the question that what will be the probability if she does not replace the first pen. Now here in total we have 5 pens. Now if a pen is selected at random and that pen is not replaced then the total pens that is the remaining pens will be 4. Now earlier we have total writing tools is equal to 12. Now in this case where a pen is selected and not replaced then the total writing tools will be 11 that is 7 pens and 4 pens. Now in the second draw a pen is selected from the remaining 11 writing tools so here you can see that the number of pens and total number of writing tools will change in second draw. So these two events are dependent events. Now here let event A be joined a pen in a first draw and event B is drawing a pen in a second draw and here A and B are dependent events. So here in the first draw probability of drawing a pen that is probability of event A is equal to. Now in the first draw we have 5 pens 12 writing tools. So probability of event A is 5 upon 12 second draw probability of event B following the event A is equal to. Now in the second draw total pens are 4 and total writing tools are 11. So probability of event B following the event A is 4 upon 11. Now using the second result which is given in the key idea we can find the probability of two dependent events. So here probability of event A and B is equal to probability of event A which is 5 upon 12 into probability of event B following the event A which is 4 upon 11. Now 4 into 3 is 12 so this is equal to 5 upon 33. So the probability that each writing tool selected is a pen if the first pen is not replaced is 5 upon 33 and this is the solution of the given question. That's all for this session hope you all have enjoyed the session.