Super Simple Explanation of Probability Tree Diagram!

[00:00:00]Today, we're going to learn a probability tree diagram. Imagine a spinner that has six equal sections. Two sections are colored blue, while the remaining four sections are colored red.

[00:00:12]A student named Jake spins this spinner twice. Our first goal is to simply list every possible outcome that can happen after the two spins. Since Jake spins the spinner two times, each spin has only two possible outcomes. The spinner can land on blue or red during the first spin, and once again on blue or red during the second spin. If we write every possible combination, we get four outcomes in total.

[00:00:43]These are blue followed by blue, blue followed by red, red followed by blue, and finally, red followed by red.

[00:00:53]Writing all the outcomes like this is manageable for a small example, but as the number of events increases, the list quickly becomes long and confusing.

[00:01:04]This is exactly why we use a probability tree diagram. Instead of writing every possibility separately, we organize them into branches so that every possible outcome can be seen clearly. We always begin with a single starting point.

[00:01:21]Since the first spin has only two possible outcomes, we draw two branches coming out from this point. One branch represents blue, while the other branch represents red.

[00:01:34]These two branches account for every possible result of the first spin.

[00:01:39]Now, we move to the second spin.

[00:01:42]No matter what happened on the first spin, the second spin can again result in either blue or red. Therefore, from each existing branch, we draw two more branches. One branch represents blue, and the other represents red.

[00:01:59]If we now trace every complete path from left to right, we obtain exactly the same four outcomes we listed earlier.

[00:02:07]The first path represents blue followed by blue.

[00:02:12]The second path represents blue followed by red.

[00:02:16]The third path represents red followed by blue.

[00:02:19]The final path represents red followed by red.

[00:02:23]At this point, the tree diagram shows every possible outcome, but it does not yet include any probabilities.

[00:02:32]To add those, we first look at the spinner.

[00:02:35]Since two of the six sections are blue, the probability of landing on blue is two out of six, which simplifies to one out of three.

[00:02:45]The remaining four sections are red, so the probability of landing on red is four out of six, which simplifies to two out of three.

[00:02:55]We now write these probabilities beside each branch of the tree.

[00:02:59]Every branch representing blue is labeled one out of three, while every branch representing red is labeled two out of three.

[00:03:08]Since the same spinner is used again for the second spin, these probabilities remain exactly the same on every second level branch.

[00:03:18]Our probability tree diagram is now complete.

[00:03:22]It shows every possible outcome together with the probability of taking each branch.

[00:03:29]Now, let's look at another example where we'll use a probability tree diagram to calculate probabilities.

[00:03:36]A student named Emily is going to roll a fair six-sided die and then flip a fair coin.

[00:03:43]She wins a prize only if she rolls a four on the die and then gets heads on the coin toss. Our task is to complete probability tree diagram.

[00:03:54]Okay, we begin by looking at the die.

[00:03:58]There are only two possible outcomes that matter for this question.

[00:04:02]Emily either rolls a four or she does not roll a four.

[00:04:07]Since there is only one face showing four on a fair six-sided die, the probability of rolling a four is one out of six.

[00:04:17]The remaining five faces are not four.

[00:04:20]So, the probability of not rolling a four is five out of six.

[00:04:24]We write these probabilities on the first pair of branches. Together, they add up to one as they always should.

[00:04:32]Now, we move to the second stage of the tree diagram, which represents the coin toss.

[00:04:38]From each of the two branches that we have already drawn, we again draw two new branches.

[00:04:44]One branch represents heads, while the other represents tails.

[00:04:49]This is because after either rolling a four or not rolling a four, the coin can still land on either heads or tails.

[00:04:57]A fair coin has two equally likely outcomes, heads and tails.

[00:05:03]Therefore, the probability of getting heads is one out of two. And the probability of getting tails is also one out of two.

[00:05:11]We write these probabilities on both sets of second level branches because the coin is fair regardless of the result of the die.

[00:05:19]Our probability tree diagram is now complete.

[00:05:23]The next step is to use this tree diagram to calculate the probability of different outcomes.

[00:05:29]Suppose we want to calculate the probability that Emily wins the prize.

[00:05:34]According to the question, she wins only if she first rolls a four and then gets heads.

[00:05:41]We begin by locating this route on the tree diagram.

[00:05:45]First, we follow the branch labeled four, and then we follow the branch labeled heads.

[00:05:51]Whenever we want to find the probability of one complete route through a probability tree, we multiply the probabilities along that route. So, in this example, we will multiply 1 by 6 and 1/2.

[00:06:05]This is the fundamental rule that is used in every probability tree question.

[00:06:11]This gives us 1 by 12.

[00:06:13]Therefore, the probability that Emily wins the prize is 1 out of 12. Amazing.

[00:06:20]Now, let's look at one last example involving probability tree diagrams.

[00:06:26]A bag contains four blue marbles and six yellow marbles.

[00:06:30]A student named Michael randomly picks one marble from the bag, replaces it, and then picks another marble.

[00:06:38]We need to complete the probability tree diagram and answer a few probability questions.

[00:06:44]Since there are two possible outcomes for the first pick, blue or yellow, we begin by drawing two branches from the starting point.

[00:06:53]Now, we move to the second pick. Because the second pick can again result in either blue or yellow, we draw two more branches from each of the first branches.

[00:07:03]This gives us all the possible outcomes for the two picks.

[00:07:07]Now, the first step is to determine the probabilities for the first pick.

[00:07:13]Since there are four blue marbles and six yellow marbles, there are 10 marbles all together.

[00:07:19]Therefore, the probability of picking a blue marble is four out of 10, while the probability of picking a yellow marble is six out of 10.

[00:07:29]These probabilities are written on the first pair of branches.

[00:07:32]Then we write the probabilities for the second pick.

[00:07:36]Since the first marble is placed back into the bag before the next pick, the contents of the bag remain exactly the same.

[00:07:44]Therefore, the probability of picking a blue marble is still four out of 10, while the probability of picking a yellow marble is still six out of 10.

[00:07:54]We write these probabilities beside each of the second-level branches, and our probability tree diagram is now complete.

[00:08:02]Next, suppose we are asked to calculate the probability that both marbles selected are blue.

[00:08:09]We first identify the correct route on the tree diagram.

[00:08:14]This route follows the blue branch on the first pick, and then the blue branch on the second pick.

[00:08:20]Since we are following one complete route, we multiply the probabilities along that path.

[00:08:27]The probability of blue followed by blue is four out of 10 multiplied by four out of 10.

[00:08:34]Therefore, the probability of selecting two blue marbles is 16 out of 100, which is 16%.

[00:08:42]Now, here's how this question becomes more interesting and fun. Suppose we are asked to calculate the probability of selecting exactly one blue marble.

[00:08:53]There are two different ways this can happen. The first possibility is selecting blue on the first pick and yellow on the second pick.

[00:09:02]The second possibility is selecting yellow on the first pick and blue on the second pick.

[00:09:08]Let us calculate the probability for the first route.

[00:09:12]The probability of blue followed by yellow is four out of 10 multiplied by six out of 10, which is 24 out of 100.

[00:09:21]Now, consider the second possible route.

[00:09:24]The probability of yellow followed by blue is six out of 10 multiplied by four out of 10, which is 24 out of 100.

[00:09:34]Since both of these routes satisfy the condition of selecting exactly one blue marble.

[00:09:39]We must add their probabilities together.

[00:09:42]24 by 100 plus 24 by 100 equals 48 out of 100.

[00:09:49]Notice that this time we are adding the probabilities because there is more than one successful route.

[00:09:56]Whenever multiple different paths satisfy the required condition, we add their probabilities together.

[00:10:04]Finally, suppose we are asked to calculate the probability that both selected marbles are the same color.

[00:10:11]There are again two successful routes.

[00:10:14]The first route is blue followed by blue, while the second route is yellow followed by yellow.

[00:10:21]Since both routes satisfy the condition, we calculate each one separately.

[00:10:27]We already know that the probability of blue followed by blue is 16 out of 100.

[00:10:34]The probability of yellow followed by yellow is six out of 10 multiplied by six out of 10 or 36 out of 100.

[00:10:43]Since both routes produce marbles of the same color, we add their probabilities together.

[00:10:49]16 by 100 plus 36 by 100 equals 52 out of 100.

[00:10:56]This is the probability that both selected marbles have the same color, and that's how probability tree diagrams work.

[00:11:05]Just remember the two key rules.

[00:11:08]Multiply probabilities along a single path, and add the probabilities of different successful paths. If you enjoyed this video, please don't forget to like, share, and subscribe to our channel.

[00:11:21]Also, you can support my channel by joining our community and becoming a member. So good.

[00:11:33]>> Mhm.