Probability of Complementary Events Sample Space

let's say we want to roll 6-sided die and so let's write the sample space for that so we have six possible outcomes we can get any number between one and six now let's say that event includes the outcomes all natural numbers so less than or equal to four an event has the outcomes three four and five given us information what is the probability of event and event occurring what do you think the answer is to this question in order to find out the first thing we need to do is write the sample space of events and so this is the intersection of and so the numbers that corresponds to event and are three and four now the probability is equal to the number of favorable outcomes divided by the total number of possible outcomes so we have two numbers where and will both be occurring out of six potential outcomes so the probability is going to be two over six which we can reduce that to as 2 times 1/6 is three times two and so this becomes probability of 1 over 3 which has decimal that's point 33 or 33% that's the rounded answer now what about the probability of event or event occurring what's the answer to that question let's write the sample space of or so this is going to be the union of these two events so basically it's going to be all these numbers combined event occurs anytime we get 1 2 3 or 4 occurs if we get 3 4 or 5 so this will include everything except 6 so this 5 favorable outcomes that lead us to get an event or out of total of six possible outcomes so the probability is 5 over 6 which as decimal that's approximately point eighty three the three is repeating so that's about eighty three point three percent chance of getting or now what about the complement of what is the probability of getting complimentary event well first we need to write the sample space of the complement of so this includes everything that is not so out of our sample space of one to six is one to four so the other numbers that are not in but still in our total sample space are the numbers of five and six so the probability of getting the complement of is going to be two you have two favorable outcomes out of total of six which is going to be run over three now what about the probability of get an event or of event occurring what would you say now there's four favorable outcomes that lead to event out of six so forma six reduces two over three now there's something that you want to take notice of notice that the probability of even or current plus the probability of the complement of vete occur adds up to 1 of is 2 over 3 and of the complement of is 1 over 3 that is 3 over 3 which is 1 so this tells us that if we wish to calculate the probability of complement or event it's 1 minus the probability of that event occurring make sure you write this equation out because it's gonna be useful in the future let's work on this problem if the probability of selecting red marble in bag full of marbles is 35% what is the probability of selecting marble that is not red so the probability of getting red marble we're told is 35% which is 0.35 so what is the probability of getting marble that is not red or the complements we have are based on the formula that we wrote before it's going to be 1 minus the probability of getting red marble so it's 1 minus 0.35 which is 0.6 5 so there's 65% chance of getting marble that is not red