Introduction to Probability Basic Overview Sample Space Tree Diagrams

in this lesson we're going to talk about probability so what is probability for example perhaps you've seen something like this of what does that mean this is the probability of event occurring to calculate the probability of an event current it's equal to the number of favorable outcomes or outcomes that lead to event and current divided by the total possible number of outcomes now before we go over some examples that talk about how to calculate probability we need to talk about something called sample space so what is sample space the sample space is basically the set of all possible outcomes that can occur so let's say if we toss fair coin let's say quarter what are the possible outcomes of flipping onecoin there's only two possibilities you can either get heads or you can get tails so the sample space for this situation is either heads or tails now what if we wanted to flip let's say two coins what are the possible outcomes what's the sample space for flipping two coins to help us get the answer we're going to create something known as tree diagram so when flipping the first coin we have two possibilities heads or tails now let's say if we get heads during the first flip during the second flip we can get another two possibilities heads or tails likewise if we get tails during the first flip on the second flip we can also get heads or tails so notice that we have four possible outcomes so the sample space is going to be the first outcome which is heads and then heads so we can write that as hh the second outcome is heads and then tails so that's going to be ht the third outcome is tails then heads and the fourth outcome is going to be tails and then tails so this would be the sample space of flipping two coins now for the sake of practice what is the sample space of flipping three coins feel free to pause the video and try that construct the tree diagram to help we do so so on the first try we can get heads or tails now if we get heads it can be heads or tails and if we get tails it can also be head toward cells now because we're flipping three coins we need to do this one more time so this could be or and then repeat the process for each one so here's the first possibility we can get three heads so i'm going to write that as hh the second possibility is getting heads heads and then tails so the third possibility is heads tails heads so that's the fourth is heads tails tails so htt and then repeat in this process we see that the next one is going to be and then tht tth and the last one is going to be so we're flipping two coins three times so the number of possible outcomes is two raised to the third power which is two times two times two and so it gives us eight possible outcomes so this is the sample space represents all the possible outcomes that we can get for flipping three coins now the probability of an event occurring is always between zero and one if the probability is 0 this means that this event cannot happen it will never happen now if the probability is equal to 1 that means that the event will always happen it also means that it has 100 chance of current if the probability of an event current is let's say 0.3 0.3 times 100 is 30 so it means that it has 30 chance of recurrent so if the probability is 0.3 that means out of let's say 10 possible tries we're gonna get approximately three favorable outcomes because three out of ten three out of ten is point three let's say out of hundred tries we would get thirty favorable outcomes so here's an example situation let's say that the probability of people who drive blue car is let's say if you have certain population city and if you randomly select person the probability of that person driving blue car let's say it's 0.20 so that means that there's 20 chance of selecting person driving blue car so if you were to randomly select 100 people 20 people would drive blue car if you randomly select thousand people approximately 200 will be driving blue car and so that's what probability tells you but now let's work on some problems if two fair coins are flipped what is the probability of getting at least one head well let's begin by writing out the sample space for flipping two coins so it could be heads heads heads tails tails heads or tnt now the event is getting at least one head so this one has at least one this one too and that one as well so the reduced sample space for is hh ht and th so now let's calculate the probability in order for the event to occur we have three favorable outcomes the total possible outcomes are four there's four events in the sample space so the probability of getting at least one heads when flipping two fair coins is going to be three over four three divided by four is point seven five which means that there's seventy 75 chance for this event occurring now let's move on to part by the way if you want to try it feel free to pause the video if three coins are flipped what is forgot the word the what is the probability of getting at least two tails so let's begin by writing out the sample space so let's write what we had before it could be th and so forth so those are the eight possible events that or outcomes that can occur in this event so now let's call the event we want to get at least two tails so which of these outcomes contains at least two tails we have one two three four so there's four potential outcomes that have at least three tails mean four yeah two tails kind of mix my words up there so let's write it out it could be http or so the number of favorable outcomes is four out of 8 possible outcomes so 4 over 8 you could reduce that to 8 is basically 4 times 2 4 is 4 times 1 so this becomes 1 over 2. 1 divided by 2 is 0.5 so there's 50 chance of getting at least two tails now let's move on to part if three coins are flipped what is the probability of getting exactly one tail so go ahead and take minute and work on this example pause the video if you want to so let's circle which outcomes or let's circle the outcomes that contain exactly one tail so this is one of them here's the other this is another one and that is it so for event or let's call it event for part the three favorable outcomes are hht hth and thh so the probability is going to be three favorable outcomes out of total of eight possible outcomes three divided by eight as decimal is point three seven five and if we multiply that by hundred that means that there's thirty seven point five percent chance of getting exactly one tail if three coins are flipped so those are some simple examples of how you can calculate the probability of an event occurring now let's move on to our second problem six sided die is tossed what is the probability of getting two let's begin by listing the sample space so here's all the possible numbers that we can get it's basically one through six now the probability of getting two is just there's only one two out of six possible outcomes so we have one favorable outcome out of six and one over six as decimal is basically 0.16 repeating if we multiply that by 100 but first let's round that to 0.167 so this is approximately 16.7 so that's the probability of getting 2 tossing 6 sided now what about part what is the probability of getting three or five so we have two favorable outcomes out of six so it's going to be two over six which if you divide both numbers by two you can reduce that to one over three so this is basically point three repeating so that's thirty three point three percent chance of getting 3 or 5. now what about part what is the probability of getting number that is at most four so let's make let's list out the outcomes that leads us to this particular event we'll call it event so that is at most four what does that mean so that means we can get numbers that is less than or equal to four so one two three four so the probability of this event occurring we could say let's see if is variable so has to be less than or equal to four that means getting number between one and four where is an integer or technically let's say it's natural number because integers can be negative so there's four favorable outcomes over six four is basically two times two six is two times three so this becomes two over three which is approximately 0.667 so there's 66.7 percent chance of event occurring now what about part what is the probability of getting number that is greater than three so let's list out the outcomes that favors event so numbers that are greater than three that includes four five and six but it does include three so the probability of getting natural number that is not equal to three but greater than three is going to be we have three favorable outcomes out of six six is three times two three is three times one so this becomes one over two which means we have 0.5 chance or 50 chance of event occurring so as you can see it's not difficult to calculate the probability of an event occurring it's pretty straightforward but with many examples you can see what to do now what about the last part part what is the probability of getting number that is less than or equal to five so event numbers that is less than or equal to five that's everything except six so the probability of getting number or natural number that is less than or equal to five it's going to be five favorable outcomes out of six potential outcomes so 5 over 6 is 0.833 repeating so let's put approximately so there's 83.3 percent chance of event occurring so that's basically it for this video now you know how to calculate the probability of an event occurring thanks for watching by the way feel free to check out my playlist statistics playlist do have more videos on probability if you are looking for those topics such as independent dependent events mutually exclusive events conditional probability and other stuff like that contingency tables and complementary events so feel free to take look at that or you could do youtube search and type in that organic chemistry tutor let's say conditional probability and it will come up thanks for watching