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Least probability: 0.70
Greatest probability: 0.80
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
26% (01:50) correct
74%
(02:26)
wrong
based on 467
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According to sports analysts, the New York Yankees have an 80% chance of winning the World Series in baseball, whereas the NY Jets have a 90% chance of winning the Superbowl.
Select for Least probability the least probability, compatible with the probabilities provided by the respective sports analysts, that both the New York Yankees and the NY Jets will win the World Series and Superbowl titles, and select for Greatest probability the greatest probability, jointly consistent with the probabilities provided by the respective sports analysts, that both the New York Yankees and the NY Jets will win the World Series and Superbowl titles. Make only two selections, one in each column.
Select for Least probability the least probability, compatible with the probabilities provided by the respective sports analysts, that both the New York Yankees and the NY Jets will win the World Series and Superbowl titles, and select for Greatest probability the greatest probability, jointly consistent with the probabilities provided by the respective sports analysts, that both the New York Yankees and the NY Jets will win the World Series and Superbowl titles. Make only two selections, one in each column.
| Least probability | Greatest probability | |
| 0.02 | ||
| 0.10 | ||
| 0.70 | ||
| 0.72 | ||
| 0.80 | ||
| 0.90 |
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Official Answer
Least probability: 0.70
Greatest probability: 0.80
31 Oct 2024, 12:00
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1. We are given the probabilities of 2 events and are the asked to find the least and greatest probabilities of them both happening.
2. This is quite confusing to think about, so let’s try to draw it out. Probability can be thought of as sets. Let there be a box and point is randomly chosen.
3. The probability that the New York Yankees will win is 80% or 80% of the area in the box. The same goes for the probability that the NY Jets will win (90%).
4. Both of these probabilities are somehow merged together. Our question can be rephrased into finding the maximum and minimum probability of picking a point in both the green and blue areas.
5. The minimum probability happens when the two areas are as separated as possible.
In this case, it’s 70% or 0.70.
6. The maximum probability happens when the two areas are as merged as possible.
In this case, it’s 80% or 0.80.
7. Our answer will be: Least probability = 0.70 and Greatest probability = 0.80.
SportsArea1.jpg [ 26.68 KiB | Viewed 3708 times ]
SportArea2.jpg [ 39.38 KiB | Viewed 3679 times ]
SportsArea3.jpg [ 34.04 KiB | Viewed 3644 times ]
2. This is quite confusing to think about, so let’s try to draw it out. Probability can be thought of as sets. Let there be a box and point is randomly chosen.
3. The probability that the New York Yankees will win is 80% or 80% of the area in the box. The same goes for the probability that the NY Jets will win (90%).
4. Both of these probabilities are somehow merged together. Our question can be rephrased into finding the maximum and minimum probability of picking a point in both the green and blue areas.
5. The minimum probability happens when the two areas are as separated as possible.
In this case, it’s 70% or 0.70.
6. The maximum probability happens when the two areas are as merged as possible.
In this case, it’s 80% or 0.80.
7. Our answer will be: Least probability = 0.70 and Greatest probability = 0.80.
Attachment:
SportsArea1.jpg [ 26.68 KiB | Viewed 3708 times ]
Attachment:
SportArea2.jpg [ 39.38 KiB | Viewed 3679 times ]
Attachment:
SportsArea3.jpg [ 34.04 KiB | Viewed 3644 times ]
26 Nov 2024, 20:19
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This is a hidden overlapping sets question, so I'll use the 2-dimension matrix to illustrate:
Because a team either wins or loses, so the total probability is 1. Therefore, the probability that Yankees loses is 1 - 0.8 = 0.2, Jet loses is 1 - 0.9 = 0.1
The question asks to find the least and the greatest probability that both Yankees and Jets win, which means to find min and max value of x
- To find min(x), we need to find max(a). Max(a) = 0.2 (in this case, b = 0). So min(x) will be 0.9 - 0.2 = 0.7
- To find max(x), let's try a = 0 or c = 0
Therefore, the least probability that both Yankees and Jets win is 0.7 and the greatest probability is 0.8
| Yankees wins | Yankees loses | Total probability | |
| Jets wins | x | a | 0.9 |
| Jets loses | c | b | 0.1 |
| Total probability | 0.8 | 0.2 | 1 |
Because a team either wins or loses, so the total probability is 1. Therefore, the probability that Yankees loses is 1 - 0.8 = 0.2, Jet loses is 1 - 0.9 = 0.1
The question asks to find the least and the greatest probability that both Yankees and Jets win, which means to find min and max value of x
- To find min(x), we need to find max(a). Max(a) = 0.2 (in this case, b = 0). So min(x) will be 0.9 - 0.2 = 0.7
- To find max(x), let's try a = 0 or c = 0
- If a = 0 then b = 0.2, which higher than the total probability of Jet loses (0.1) => eliminate
- If c = 0 then b = 0.1, a = 0.1, x = 0.8 => compatible with the probabilities provided
Therefore, the least probability that both Yankees and Jets win is 0.7 and the greatest probability is 0.8
