Question Video: Applying the Multiplication Rule to Calculate Probabilities for Two Dependent Events Mathematics

The probability that it rains on a given day is 0.6. If it rains, the probability that a group of friends play football is 0.2. If it does not rain, the probability that they play football rises to 0.8. Work out the probability that it rains on a given day and the friends play football. Work out the probability that it does NOT rain on a given day and the friends play football. What is the probability that the friends will play football on a given day?

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Video Transcript

The probability that it rains on a given day is 0.6. If it rains, the probability that a group of friends play football is 0.2. If it does not rain, the probability that they play football rises to 0.8. Work out the probability that it rains on a given day and the friends play football. Work out the probability that it does not rain on a given day and the friends play football. What is the probability that the friends will play football on a given day?

There are three parts to this question. They all involve conditional probability and the dependent events whether it rains and whether a group of friends play football. One way to represent the information from the question is using a tree diagram. We will now clear some space to do this first.

We will begin by letting 𝑅 be the event that it rains. We are told that the probability that it rains on any given day is 0.6. We know that the complement of any event 𝐴, which is written 𝐴 prime or 𝐴 bar, has a probability that is equal to one minus the probability of 𝐴. This means that the probability it does not rain in this question is one minus 0.6. This is equal to 0.4 and can be added to the tree diagram as shown.

If we let the event that the group of friends play football be 𝐹, there are four possible scenarios: firstly, that it rains and the friends play football; secondly, that it rains and the friends do not play football; thirdly, that it does not rain and the friends play football; and finally, that it does not rain and the friends do not play football. We are told that if it rains, the probability that the friends play football is 0.2. This is an example of conditional probability, the probability that the friends play football given that it rains. We can then add 0.2 to our tree diagram.

Once again, since the probabilities on each pair of branches sum to one, the probability of the complement of this is 0.8. The probability that the friends do not play football given that it rains is 0.8. We can repeat this for the bottom half of our tree diagram. We are told in the question that if it does not rain, the probability that the friends play football is 0.8. The conditional probability that the friends play football given that it does not rain is 0.8.

Let’s now return to the three specific questions we were asked. Firstly, we were asked to work out the probability that it rains on a given day and the friends play football. As we want both events to occur, this is the intersection of the two events. We recall that given two events 𝐴 and 𝐡, the probability of 𝐴 intersection 𝐡 is equal to the probability of 𝐡 given 𝐴 multiplied by the probability of 𝐴. In this question, the probability that it rains and the friends play football is equal to the probability that the friends play football given that it rains multiplied by the probability it rains. We need to multiply the probabilities 0.2 and 0.6. This is equal to 0.12.

Let’s now consider the second part of our question. This asked us to work out the probability that it does not rain on a given day and the friends play football. This corresponds to the pink path on our tree diagram. The probability that it does not rain and the friends play football is equal to the probability they play football given that it does not rain multiplied by the probability it does not rain. We need to multiply 0.8 and 0.4. This is equal to 0.32. The probability that it does not rain on a given day and the friends play football is 0.32.

The final part of our question asked us to calculate the probability that the friends play football on a given day. This can occur in one of two ways: either it rains and they play football or it does not rain and they play football. We need to find the union of these two events. From the tree diagram, this involves finding the sum of the probabilities. We need to add 0.12 and 0.32. This is equal to 0.44. We can therefore conclude that the probability that the friends play football on a given day is 0.44.

It is worth noting that the sum of the probabilities for every possible outcome combined is equal to one. In this case, the four probabilities 0.12, 0.48, 0.32, and 0.08 sum to one.

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