take a look at an example you have a well-shuffled deck of 52 cards you draw two cards from the deck find the probability that the two cards you draw are both Queens if you draw the cards one without width replacement or two without replacement so let's start with number one here okay actually let's start with defining our events so we'll call q1 the event where we draw the first queen the first card is a queen and Q2 the event where the second card is a queen now let's start with number one with replacement now remember when we do this with replacement that means the two events are going to be independent okay so if we want to know the probability that we draw a queen and then another queen that's going to be equal to the probability q1 times the probability of Q2 this makes the calculation considerably easier as you'll see okay q1 and Q2 we just calculate these two probabilities so separately this is because again we're doing this with replacement which means the two events are independent all right so let's take it one at a time what's the probability that we draw that first so you reach in the deck 52 cards and you pull out one of these Queens okay now there are four Queens in the deck of 52 cards so the chance that you draw a queen is 4 out of 52. notice we're dealing with the deck of cards so we're again talking about outcomes that are equally likely so we just have to count how many queens we have divide by all the possibilities okay then what and when we're doing this with replace so we pull that Queen out we look at and say oh it's a queen and then what do we do we replace it back in the deck after reshuffling we draw another card now what's the probability of drawing a queen well because we put the queen back it's back in the deck which means there are still Four Queens to choose from out of 52 cards total so there are four Queens to choose from out of 52 card so for that second draw okay bring over our calculator that means we have 4 out of 52 times 4 out of 52. the probability of drawing two queens with replacement is 0.0059 that's not a big chance right that's less than one percent chance okay now let's see what happens if we do this without replacement now this makes things a little more complicated because now the two events are not going to be independent so when we calculate the probability that we draw two queens that is the probability that we draw one Queen and then another queen we first calculate the probability of drawing that first queen but then when we calculate the probability of drawing the second Queen we have to take into account that we've already taken a queen out of the deck so probability of drawing that first queen is the it's exactly the same as in the previous part for that first draw nothing has changed we saw four coins in the deck out of 52 cards total so Four Queens to choose from out of 52 card Soto so I reach my hand in I draw a card it's a queen in fact it's the queen of hearts and then I don't put it back I keep it here in my hand okay so now that queen of hearts since I do not replace it back in the deck it's not there anymore sorry that's the diamonds let's do hearts that Queen of Hearts isn't there in the deck anymore okay so I need to take that into account when I do the second draw so taking into a fact that the that Queen is now gone from the deck what's the probability that I draw that second Queen well now I do it the same way but now with that first coin gone there's only three queens left in the deck right one of the Queens is out of the deck the deck has changed now there's only three coins left so there's three queens to choose from now and I don't have 52 cards left in my deck because I took out that Queen so I have one less card than I started with now I only have 51 cards to choose from for that second draw okay and so that is how we calculate the probability of drawing two queens without replacement let's see what the calculator says about it okay so we have 4 out of 52 because for that first draw there are four Queens out of 52 cards total but then I don't put it back so taking into account that extra information because I don't put the queen back now I only have three queens to choose from out of 51 total cards and so the probability of drawing two coins without replacement is just a little bit less 0.0045 still less than one percent but you could see that they're a little different