Back to Edexcel Probability (H) Home
5 C) Working Out Probability – Part 2
5 C) Working Out Probability – Part 2
Mutually Exclusive
Some events are mutually exclusive, which means that the events cannot occur at the same time. For example, the two outcomes when we flip a coin are heads and tails. Only one of these outcomes can occur; we cannot flip a coin and get a head and a tail.
Another example would be if we had a bag with red and blue counters in. When we reach into the bag and pull out a counter, the counter will be either red or blue. We cannot get a counter that is a mixture of both of the colours.
The final example that we are going to look at is the outcome of a football match. The three outcomes from a football match is win, draw or lose, and the outcome of this game can only be one of the three options and never more than one; you cannot lose and draw a football.
The probability of mutually exclusive events adds up to 1. This means that if we are given the probabilities for all bar one of mutually exclusive events occurring, we are able to work out the probability of the missing event.
Some events are mutually exclusive, which means that the events cannot occur at the same time. For example, the two outcomes when we flip a coin are heads and tails. Only one of these outcomes can occur; we cannot flip a coin and get a head and a tail.
Another example would be if we had a bag with red and blue counters in. When we reach into the bag and pull out a counter, the counter will be either red or blue. We cannot get a counter that is a mixture of both of the colours.
The final example that we are going to look at is the outcome of a football match. The three outcomes from a football match is win, draw or lose, and the outcome of this game can only be one of the three options and never more than one; you cannot lose and draw a football.
The probability of mutually exclusive events adds up to 1. This means that if we are given the probabilities for all bar one of mutually exclusive events occurring, we are able to work out the probability of the missing event.
Example 1
Suppose that we have a bag that contains 3 coloured balls; green, blue and red. The probability of picking a green ball is 0.3 and the probability of picking a blue ball is 0.2. What is the probability of picking a red ball?
These events are mutually exclusive; only one of the colour balls can be picked. This means that the sum of the probabilities of all of these events occurring must equal 1.
Suppose that we have a bag that contains 3 coloured balls; green, blue and red. The probability of picking a green ball is 0.3 and the probability of picking a blue ball is 0.2. What is the probability of picking a red ball?
These events are mutually exclusive; only one of the colour balls can be picked. This means that the sum of the probabilities of all of these events occurring must equal 1.
We are given the probability of picking a green ball (0.3) and the probability of picking a blue ball (0.2). We can sub these values into the above equation.
The first step in finding out the probability of picking a red ball is to simplify the left side of the equation; we collect the two numbers.
We now need to move the 0.5 that is on the left to the right side of the equation and we are able to do this by doing the opposite; we take 0.5 from both sides of the equation.
Therefore, the probability of picking a red ball is 0.5.
Example 2
The probability that a football team will draw is 0.1. The team are twice as likely to lose than win. What is the probability that the team will lose? And, what is the probability that the team will win?
Wining, losing and drawing are all mutually exclusive events. Therefore, the sum of the probabilities of these events occurring will be equal to 1.
The probability that a football team will draw is 0.1. The team are twice as likely to lose than win. What is the probability that the team will lose? And, what is the probability that the team will win?
Wining, losing and drawing are all mutually exclusive events. Therefore, the sum of the probabilities of these events occurring will be equal to 1.
We are given the probability that the team will draw in the question, which is 0.1. So, let’s sub this value in to the above equation.
The question also tells us that it is twice as likely that the team will lose than win. Let’s let the probability of the team winning equal x. As the team is twice as likely to lose, we can represent the probability that the team will lose as 2x. We can sub x in as the probability of the team winning and 2x in as the probability of the team loosing.
The first step in finding the value of x is to collect the two terms on the left side of the equation that contain x’s.
We now need to get all of the x terms on one side of the equation and all of the numbers on the other side of the equation. There are more x’s on the left side of the equation than the right. Therefore, I am going to get all of x’s to the left and all of the numbers to the right. This means that I need to move the 0.1 that is currently on the left to the right. We are able to do this by doing the opposite; we take 0.1 from both sides of the equation.
We now have all of the x’s on one side and all of the number on the other side. We want to find the value of x and not 3x. This means that we need to divide both sides by 3 (we divide both sides by the coefficient of the unknown).
Therefore, x is 0.3. We let x equal the probability of the team winning and this means that the probability that the team will win is 0.3.
The probability that the team will lose is twice the probability that the team will win (2x). This means that we can work out the probability that the team will lose by multiplying the probability of the team winning by 2, which gives us a value of 0.6 (2 x 0.3).
It is always good to check that we have the correct values for the probabilities. We can do this by subbing the probabilities that we have found and checking that they add up to 1 (they should all add up to 1 because the three events are mutually exclusive).
The probability that the team will lose is twice the probability that the team will win (2x). This means that we can work out the probability that the team will lose by multiplying the probability of the team winning by 2, which gives us a value of 0.6 (2 x 0.3).
It is always good to check that we have the correct values for the probabilities. We can do this by subbing the probabilities that we have found and checking that they add up to 1 (they should all add up to 1 because the three events are mutually exclusive).
The equation above holds, which means that the probability that the teams wins is 0.3 and the probability that the team loses is 0.6.
Example 3
Sometimes it will be the case that we are given the probabilities of different mutually exclusive events occurring in a table form. We will usually be asked to find the probabilities for the gaps in the table. We are able to answer questions like this because we know that the probabilities of all of the events will add up to 1. Let’s now have an example.
A spinner has 3 different outcomes; A, B and C. Fill in the table.
Sometimes it will be the case that we are given the probabilities of different mutually exclusive events occurring in a table form. We will usually be asked to find the probabilities for the gaps in the table. We are able to answer questions like this because we know that the probabilities of all of the events will add up to 1. Let’s now have an example.
A spinner has 3 different outcomes; A, B and C. Fill in the table.
The events on the spinner will be mutually exclusive, which means that they cannot occur at the same time as each other. Therefore, the probabilities of the three different outcomes will add up to 1. This gives us the equation below.
We can now sub the values in from the table.
We can collect the two numbers together on the left side of the equation.
The next step is to move the 0.8 from the left side of the equation to the right and we do this by taking 0.8 from both sides of the equation.
Therefore, the probability of the spinner landing on a B is 0.2. the completed table is shown below.