12 7 Probability of Compound Events

section 127 is on the probability of compound events in this lesson we're going to take look at four different situations the first is the probability of independent events if the probability of one event occurring is not dependent on the other event then to find the probability of both events occurring multiply the probabilities so the probability of and happening is equal to the probability of happening times the probability of let's take look at an example here so we have bag that contains three red five blue and six yellow marbles marble is selected replaced and then second marble is selected we need to find each probability so the first is what is the probability of drawing red marble and then Blue Marble so that's going to be the probability of drawing red times the probability of drawing blue now the probability of drawing red is three out of the total number of marbles which is 14 times the probability of blue five out of 14 now just put that into your calculator you can do actually the easiest way to do it is 3 divided 14 * 5 / 14 or if you just want to do it the old fashioned way of multiplying across that would be 15 over 196 punch that into your calculator you're going to get decimal 8 something which is approximately 8% second example what's the probability of drawing yellow marble and then marble that's not red so that's going to be the probability of yellow times the probability of not red so the probability of yellow marble is six out of 14 because there are six yellow Marbles and the number of marbles that are not red would be the number of blue plus yellow or 11 out of 14 and that's equal to if you do little cancelling you'll get 33 over 98 which is approximately equal to 34% here's another example so there's 30% chance of rain in Paris and 40% chance of rain in New York York what is the probability that it will rain in both Paris and New York notice that it rainning in New York is not at all impacted by whether or not it's raining in Paris those cities are far enough apart that they're two completely independent events so the prob probability of rain in Paris is 30% and so I'll write that as decimal 3 times the probability that it's going to rain in New York which is 40% and I'll write that as decimal as04 those multiply to12 which is equal to 12% so there's 12% chance that it will be reigning in both Paris and New York second one we're going to take look at now is the probability of dependent events if the probability of one event occurring is dependent on another event then multiply the probability of the first event by the probability of the second event occurring after the first so that's the probability of and occurring is equal to the probability of times the probability of occurring following let's look at an example so we're going to look at the same bag of marbles we had three red five blue and six yellow but this time once we draw marble out it is not replaced remember in the previous example we put the marble back in the bag so our total number of marbles always remained at 14 now once take marble out I'm not going to put it back in there so we need to find each probability so the probability of drawing out red and then yellow so the probability of drawing red is three out of 14 okay and now that I've taken red marble out of the bag there are now 13 marbles in the bag so now the probability of yellow is six out of 13 that's equal to 9 over 91 or approximately 10 % chance notice that when don't replace the marble the total number of marbles in the bag is decreased by one each time example number two what's the probability of drawing blue and then red and then yellow marble so the probability of Blue Marble is 5 out of 14 the probability of red is going to be three out of 13 because there's one fewer marbles in the bag and the probability of yellow is going to be 6 out of 12 12 punching that all into your calculator you get approximately 4% example three what's the probability of drawing Blue Marble another Blue Marble and then red so the first blue is going to be five out of 14 because there's five blue marbles in total of 14 now the second is clearly going to be out of 13 CU didn't put that first Blue Marble back but I've already taken one Blue Marble out of the bag so that means there are only four left in the bag and then that's going to be times red which is going to be three over 12 remember continue to decrease your denominator by one each time you draw out marble and that's approximately equal to 3% so there 3% chance that I'll draw blue blue and then red my last example here what's the probability of drawing red and then not blue so there are three three out of 14 red marbles and then how many not blue marbles are there well there's five blue so there's total of 3 + 6 or nine not blue but I've already taken red one out which is not blue so there's only eight not blue marbles left in there so eight out of 13 and that comes out to about 133% chance our next situation is the probability of mutually exclusive events to find the probability of two events that cannot occur at the same time we're going to add the two probabilities so what is the prob the probability of or happening is the probability of happening plus the probability of notice this time it's not I'm only doing one thing I'm only drawing like for example one marble or one thing happening so what's the probability of this happening or that happening here's an example die is rolled and what is the probability of rolling two or four so the probability of two or four is going to be equal to the probability of rolling two plus the probability of rolling four so the probability of rolling two is one out of six because there's only one two on six-sided die plus the prob probability of rolling four which is also 1 out of six which is equal to 2 six 13 or approximately 33% here's another example card is randomly chosen from standard deck of cards find each probability so what's the probability of drawing three or four so that's going to be the probability of drawing three plus the probability of drawing four because those are two exclusive events they cannot occur at the same time probability of rolling of drawing three is four out of 52 because there are four threes out of total of 52 cards in the deck plus same thing for four there are four fours in the deck out of total of 52 that's equal to 8 over 52 or 2 over 13 or approximately 15% okay second example what's the probability of drawing an ace or face card so that's going to be the probability of drawing an Ace Plus the probability of drawing face card and again these two events cannot occur at the same time card cannot be both an Ace and face card at the same time so the probability of drawing an ace is four out of 52 because there are four aces and the probability of drawing face card is 12 out of 52 now remember the face cards are the kings queens and Jacks and there's king queen and Jack of each of the suits hearts diamonds Spades and clubs so that's total of 12 face cards so that's 16 out of 52 four out of 13 or approximately 31% chance of drawing an ace or face card and the last situation we're going to take look at is the probability of mutually inclusive events so to find the probability of two events that can occur at the same time we're going to add the two probabilities then subtract the probability of both occurring at the same time so the probability of or occurring is going to be the probability of plus the probability of minus the probability of and and let's take look at some examples so again we have standard deck of cards and we're going to choose one card out of that deck find each probability so what is the probability of drawing heart or queen so that's going to be the probability of drawing heart plus the probability of drawing queen and now we need to ask ourselves can card be both heart and and queen at the same time and indeed it can there is one queen of hearts in the deck so need to subtract the probability of drawing the Queen of Hearts okay so there are 13 out of 52 hearts in the deck plus four Queens in the deck and there is one Queen of Hearts so need to subtract the Queen of Hearts so that's going to give me 16 over 52 which reduces to 4 out of 13 which is approximately 31% so you're probably wondering why do we have to subtract here because the Queen of Hearts is actually when when I'm talking about these 13 cards here that are Hearts the Queen of Hearts is one of those then when I'm talking about these four cards here that are Queens the Queen of Hearts is one of those can't count it twice can only count it once so have to subtract one of those occurrences of the Queen of Hearts being in the probability and let's take look at one more example here what's the probability of drawing black card or king so we need the probability of black card plus the probability of king minus the probability of black King okay so in the deck there are 26 black cards that would be 13 clubs and Spades there are four Kings okay and of those Four Kings there are two black kings that would be the King of Spades and the King of Clubs so that's going to give me 28 out of 52 which reduces to 7 over 13 or approximately 54% and again when I'm talking about my counting all of my black cards here there's King of Spades and king of clubs in there when I'm counting my Kings here have King of Spades and King of Clubs and can't count them in both places so that's why have to subtract two black kings from this so in this lesson we took look at several different types of compound probability of compound events of independent events of dependent events and of mutually exclusive and mutually inclusive events the first two examples what we talked about notice they use the word and and anytime you see the word and maybe I'll add this to the notes Here If I'm doing this and that I'm going to be multiplying and then notice in our second two examples of the mutually exclusive and inclusive events use the word or what's the probability of drawing queen or heart and in those cases had to add and sometimes had to subtract if both events could occur at the same time