Probability of Independent and Dependent Events 6 2

in this video we'll be talking about independent events and dependent events both of these events will be defined with respect to probability what are independent events independent events refer to the occurrence of one event not affecting the probability of another event for example let's say we are rolling die and flipping coin both of these are two separate events we can say that the first event is rolling or die and the second event is flipping coin because the outcome of the first event does not affect the outcome of the second event these events are said to be independent events in other words rolling six doesn't increase or decrease the probability of coin landing on heads or tails the probability of getting heads is 0.5 and it stays that way regardless of what you roll to calculate the probability of two independent events happening together you can use this formula where the probability of and is equal to the probability of event times the probability of event let's do an example if you roll 6-sided die and flip coin what is the probability of rolling 5 and getting heads the first thing we should do is write down the formula but in order to use this formula we need to know the probabilities of each event if you watch the previous video you should know that the probability of an event is equal to the total number of favorable outcomes divided by the total number of possible outcomes for the first event there is only one favored outcome which is rolling 5 and there are our total of six possible outcomes since we are rolling six-sided die as result the probability of rolling 5 is equal to 1 over 6 for the second event we know that the probability of getting heads is equal to 1 over 2 or 50% and we know this because there is only one desired outcome which is getting heads and there are total of two possible outcomes since the coin can land on either heads or tails now that we have the probabilities for each event we can use the formula and all we have to do is multiply them together 1 over 6 times 1 over 2 give an answer of 1 over 12 as result the probability of rolling 5 and getting heads is equal to 1 over 12 or 0.08 33 what are dependent events dependent events are simply the opposite of independent events dependent events refer to the occurrence of one event affecting the probability of another event for example suppose we have box that contains 10 marbles 7 other marbles are green and three of the marbles are blue based on this we know that the probability of drawing one green marble is 7 over 10 or 0.7 and the probability of drawing one blue marble is 3 over 10 or 0.3 if we randomly select two marbles from this box what is the probability of drawing green marble and then blue marble with our replacement common mistake in solving this problem is by using the formula and then multiplying the probabilities of each marble together so you'll have 7 over 10 times 3 over 10 however this process is incorrect this formula can only be used for independent events and we know that this is not an independent event since the marbles are being drawn without replacement the term without replacement means we are drawing the marble without putting it back into the box this means that the probability will change after every draw as result this is dependent event where the probability of one event affects the probability of another event in other words drawing the first marble affects the probability of the next marble let's see why this is for the first event there are 10 marbles in the box and since we have total of 7 green marbles the probability of drawing one green marble is 7 over 10 or 0.7 for the second event the probability of drawing blue marble is not 3 over 10 since there is total of nine marbles left in the box with total of 3 blue marbles remaining the probability of drawing blue marble is now equal to 3 over 9 or 0.33 as you can see the probability of drawing blue marble has changed at first it had value of 0.3 but now it has value of 0.33 or 3 over 9 as result this is dependent event because the occurrence of the first event affected the probability of the second event now to finish this problem all we have to do is multiply these two values together seven over ten times three over nine gives us an answer of seven over thirty or zero point two thirty three let's do another example using the same scenario what is the probability of drawing two green marbles without replacement feel free to pause the video so you can try this question for yourself to solve this question we we use the formula except we have to make some modifications to it the probability of and is equal to the probability of time's the probability of after event has occurred will assign event as drawing the first screed marble and will assign event as drawing the second green marble the probability for drawing the first green marble is equal to 7 over 10 since the box is untouched if this event was successful there will be six green marbles remaining with total of nine marbles left in the box therefore the probability of drawing the second green marble is equal to 6 over nine and finally to get the answer all we have to do is multiply these two values together 7 over 10 times 6 over 9 gives us an answer of 7 over 15 or 0.46 67 to quickly recap for independent events the outcome of one event does not affect the outcome of the other event if events and are independent the probability of and occurring is equal to the probability of time's the probability of and for dependent events the outcome of one event does influence the outcome of the other event this is commonly seen when drawing items are not returned if events and are dependent the probability of and occurring is equal to the probability of time's the probability of after event has occurred if you found this video helpful consider supporting us on patreon to help us make more videos you can also visit our website at simple earning procom to get access to many study guides and practice questions thanks for watching