Question Video: Finding the Conditional Probability of an Event from a Tree Diagram Mathematics

A bag contains 3 red marbles and 5 blue marbles. Two marbles are selected without replacement. Using a tree diagram, find the probability that the second marble is red, given that the first one is red. Give your answer to the nearest two decimal places.

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Video Transcript

A bag contains three red marbles and five blue marbles. Two marbles are selected without replacement. Using a tree diagram, find the probability that the second marble is red given that the first one is red. Give your answer to the nearest two decimal places.

There are many ways to answer this question. However, we are told to use a tree diagram. We are told that two marbles are selected without replacement. The first marble could be red or blue. As there are three red marbles out of a total of eight, the probability of selecting a red marble is three out of eight or three-eighths. In the same way, the probability that the first marble is blue is five-eighths. Next, we select a second marble, and as this is done without replacement, we are dealing with conditional probability.

If the first marble was red, we have two red marbles left in the bag. This is out of a total of seven marbles. Therefore, the probability that the second marble is red given that the first one is red is two-sevenths. If the first marble is red, the probability that the second marble is blue is five-sevenths. Repeating this for the bottom half of our tree diagram where the first marble selected is blue, the probability that the second marble is red is three-sevenths as there are three red marbles out of the seven remaining. And the probability the second marble is blue is four-sevenths as four of the seven remaining marbles are blue.

As already mentioned, we are dealing with conditional probability. And we need to find the probability that the second marble is red given that the first marble is red. From the tree diagram, we can see that this is equal to two-sevenths. This is not the final answer as we are asked to give our answer to the nearest two decimal places. Dividing two by seven gives us 0.2857 and so on. The probability that the second marble is red given that the first one is red is 0.29 to two decimal places.

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