MHB How Do Ace Distributions Affect Probabilities in a Bridge Game?

AI Thread Summary
Ace distributions in bridge significantly impact probabilities, as illustrated by various events involving players North, South, East, and West. The notation indicates whether a player has a certain number of aces, with primes denoting the absence of aces. For example, West having no aces is represented by W_1', while W_2 - W_3 indicates West has exactly two aces. The analysis concludes that West has zero aces in events W_1', N_2S_2, and (N_2 ∪ S_2)E_2, one ace in events N_1S_1E_1W_1 and N_3W_1, two aces in W_2 - W_3, and four aces in N_1'S_1'E_1'. Understanding these distributions is crucial for calculating probabilities in bridge.
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Bridge : For k= 1,2,3 ,4 let $N_k$ be the event that North has at least k aces. Let $S_k, E_k, W_k$ be be the analogous events for South, East, West. Discuss the number x of aces in West's possession in the events
a)$W_1', $
b) $N_2S_2,$

c) $N_1'S_1'E_1'$
d) $W_2- W_3$
e)$N_1S_1E_1W_1$

f) $N_3 W_1$
g)$(N_2 \cup S_2)E_2$

How to answer these questions? Truly speaking, I didn't follow these events. Would any member of math help board answer these questions along with explanation of these events?
 
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It helps to know the symbolism. In symbolic logic the prime, ', indicates that the statement is NOT true. Since we are told that [math]W_1[/math] means "West has at least one ace", [math]W_1'[/math] means "West does NOT have at least one ace". I.e. West does not have any aces.

Symilarly, two symbols written together indicate that they are both true. [math]N_2S_2[/math] means that "North has two at least aces and South has at least two aces".

The union sign, [math]\cup[/math], indicates "either or". [math](N_2\cup S_2)E_2[/math] means "Either North has at least two aces or South has at least two aces, and East has at least two aces.

Finally A- B means that statement A is true but statement B is false. [math]W_2- W_3[/math] means "West has at least two aces but West does not have at least three aces". I.e., West has exactly three aces.

Now you try the others.
 
County Boy said:
Finally A- B means that statement A is true but statement B is false. [math]W_2- W_3[/math] means "West has at least two aces but West does not have at least three aces". I.e., West has exactly three aces.

The last sentence should have been "West has exactly TWO aces".
 
County Boy said:
The last sentence should have been "West has exactly TWO aces".
The number x of aces in West's possession in the event $(a)W_1', (b)N_2S_2, (g)(N_2 \cup S_2)E_2$ is = 0.

The number of aces in West's possession in the event $(e) N_1S_1E_1W_1, (f)N_3W_1$ is = 1.

The number of aces in West's possession in the event $(d)W_2- W_3$ is = 2.

The number of aces in West's possession in the event $(c) N_1' S_1'E_1'$ is = 4.
 
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