According to sports analysts, the New York Yankees have an 80% chance

Question

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.

Bismuth83 · Accepted Answer

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.

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hienthusiast · Answer

This is a hidden overlapping sets question, so I'll use the 2-dimension matrix to illustrate:

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

cdrectenwald · Answer

Can someone explain this?

Actually I looked into this some more:

Least Probability 
The least possible probability that both events occur is calculated when the events are as mutually exclusive as possible, given their individual probabilities. Since probabilities cannot be negative, we use the formula:
Least Probability=max⁡(0,P(A)+P(B)−1)	ext{Least Probability} = \max(0, P(A) + P(B) - 1)Least Probability=max(0,P(A)+P(B)−1)
Applying the values:
Least Probability=max⁡(0,0.8+0.9−1)=max⁡(0,0.7)=0.7 or 70%	ext{Least Probability} = \max(0, 0.8 + 0.9 - 1) = \max(0, 0.7) = 0.7 	ext{ or } 70\%Least Probability=max(0,0.8+0.9−1)=max(0,0.7)=0.7 or 70%
 
 Greatest Probability 
The greatest possible probability occurs when the events are fully dependent (i.e., the occurrence of one ensures the occurrence of the other). The maximum probability is the lesser of the two individual probabilities:
Greatest Probability=min⁡(P(A),P(B))	ext{Greatest Probability} = \min(P(A), P(B))Greatest Probability=min(P(A),P(B))
Applying the values:
Greatest Probability=min⁡(0.8,0.9)=0.8 or 80%	ext{Greatest Probability} = \min(0.8, 0.9) = 0.8 	ext{ or } 80\%Greatest Probability=min(0.8,0.9)=0.8 or 80%

anshgupta433 · Answer

Bunuel   KarishmaB   bb   chetan2u   Bismuth83  Could you help explain this?

bumpbot · Answer

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According to sports analysts, the New York Yankees have an 80% chance

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Least probability	Greatest probability
		0.02
		0.10
		0.70
		0.72
		0.80
		0.90

	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

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According to sports analysts, the New York Yankees have an 80% chance

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