The table shows the probabilities of each result.
| Result | Win | Draw | Lose |
| -------- | ----- | ----- | ----- |
| Probability | | 0.05 | 0.3 |
Each win is worth 3 points.
Each draw is worth 1 point.
Each loss is worth 0 points.
The football team plays 40 games in a season.
Work out how many points the football team should receive in one season.
Answer (1)
Calculate the probability of a win: P ( Win ) = 1 − 0.05 − 0.3 = 0.65 .
Calculate the expected number of wins, draws, and losses: E ( Win s ) = 0.65 × 40 = 26 , E ( Dr a w s ) = 0.05 × 40 = 2 , E ( L osses ) = 0.3 × 40 = 12 .
Calculate the total expected points: E ( T o t a l P o in t s ) = 3 × 26 + 1 × 2 + 0 × 12 = 80 .
The football team should receive 80 points in one season.
Explanation
Understand the problem and provided data We are given the probabilities for a draw (0.05) and a loss (0.3) in a football team's games. We also know that each win is worth 3 points, each draw is worth 1 point, and each loss is worth 0 points. The team plays 40 games in a season, and we need to calculate the expected total points the team will receive.
Calculate the probability of a win First, we need to find the probability of a win. Since the probabilities of all possible outcomes (win, draw, loss) must add up to 1, we can calculate the probability of a win by subtracting the probabilities of a draw and a loss from 1: P ( Win ) = 1 − P ( Dr a w ) − P ( L oss ) P ( Win ) = 1 − 0.05 − 0.3 = 0.65
Calculate the expected number of wins, draws, and losses Next, we calculate the expected number of wins, draws, and losses in the 40 games: Expected number of wins: E ( Win s ) = P ( Win ) × N u mb er o f g am es E ( Win s ) = 0.65 × 40 = 26 Expected number of draws: E ( Dr a w s ) = P ( Dr a w ) × N u mb er o f g am es E ( Dr a w s ) = 0.05 × 40 = 2 Expected number of losses: E ( L osses ) = P ( L oss ) × N u mb er o f g am es E ( L osses ) = 0.3 × 40 = 12
Calculate the total expected points Now, we calculate the total expected points by multiplying the expected number of wins by 3, the expected number of draws by 1, and the expected number of losses by 0, and then summing the results: E ( T o t a l P o in t s ) = 3 × E ( Win s ) + 1 × E ( Dr a w s ) + 0 × E ( L osses ) E ( T o t a l P o in t s ) = 3 × 26 + 1 × 2 + 0 × 12 = 78 + 2 + 0 = 80
State the final answer Therefore, the football team is expected to receive 80 points in one season.
Examples
This type of probability calculation is useful in sports analytics for predicting team performance over a season. By knowing the likelihood of different outcomes (win, draw, loss) and the points associated with each, one can estimate a team's expected total points. This can help in setting realistic goals, making strategic decisions, and evaluating the team's success relative to expectations. For example, if a team consistently scores fewer points than expected based on their win/draw/loss probabilities, it might indicate a need to adjust their training or game strategies.
