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5 F) Predicting Outcomes – Part 2
5 F) Predicting Outcomes – Part 2
The content in this section builds on the content that was looked at in the section before. Make sure that you have gone through the content in the previous section before working through this section (click here to be taken through to the previous section).
Example 1
We have a football team and the probability that they win is 0.4, the probability that they draw is 0.1 and the probability that they lose is unknown. A season has 40 games in it.
The first part of the question asks us to work out the expected number of games that the football team loses throughout the season. To work out the expected number of games that the team will lose, we use the following formula:
We have a football team and the probability that they win is 0.4, the probability that they draw is 0.1 and the probability that they lose is unknown. A season has 40 games in it.
- What is the expected number of times that the team will lose throughout the season?
- What is the expected number of points that the team will get after the 40 games? Note: the team gets 3 points for a win, 1 for a draw and 0 for a loss.
The first part of the question asks us to work out the expected number of games that the football team loses throughout the season. To work out the expected number of games that the team will lose, we use the following formula:
We are given the number of games that are in a season, which is 40, but we do not have the probability that the team loses. However, we can use the information that we are given in the question to work out the probability that the team loses. This is because winning, losing and drawing are mutually exclusive events, which means that the team will either win, lose or draw. Two of these events cannot occur at the same time; for example, the team cannot win and draw a football game. We know that the sum of the probabilities of mutually exclusive events add up to 1. This means that the sum of the probability of winning, losing and drawing will be 1.
We have been given the probability of the team winning and the probability of the team drawing. We can sub these numbers into the above equation to find the probability of the team losing.
The first step in finding the probability of the team losing is to collect the numbers on the left side of the equation.
The next step is to move the 0.5 from the left side of the equation to the right side of the equation. We are able to do this by taking 0.5 from both sides of the equation.
Therefore, the probability that our football team will lose is 0.5. We now sub the probability and the number of games played (40) into the expected number of losses equation.
Therefore, it is expected that the team will lose 20 games throughout the 40 game season.
The second part of the question asks us to work out the expected number of points that the team will gain throughout the season. We are able to do this by finding the expected number of wins, losses, and draws and multiplying these values by the respective points gained for winning, losing and drawing.
The second part of the question asks us to work out the expected number of points that the team will gain throughout the season. We are able to do this by finding the expected number of wins, losses, and draws and multiplying these values by the respective points gained for winning, losing and drawing.
We are given the number of points that are gained from winning, losing and drawing; 3 points are gained for a win, 0 for a loss and 1 for a draw. Let’s sub these values in the above equation.
We are able to simplify the above equation. The first bit of simplifying that we can do is to get rid of the “(expect losses x 0)” and this is because anything multiplied by 0 is 0, thus meaning that this disappears. Also, the “(expected draws x 1)” will become the “expected draws” because anything multiplied by 1 is itself. The equation becomes:
The next step is to find out the number of wins and draws. We do this in exactly the same way as finding the expected number of games that the team loses. Let’s first find the number of games that the team wins, and we do this by multiplying the probability of the team winning (0.4) by the number of games that are played (40).
Let’s now find the expected number of draws. We multiply the probability of the team drawing (0.1) by the number of games played (40).
So, throughout the 40 game season, the football team is going to win 16 games, draw 4 games and lose 20 games (a good way to check the answer for this question is to check that the number of wins, draws and losses equals 40, which they do in this case; 16 + 4 + 20 = 40).
We can now find the expected number of points by subbing in the appropriate values.
We can now find the expected number of points by subbing in the appropriate values.
Therefore, the expected number of points that the teams gains throughout the season is 52.