Basic Probabilities. Conditional Prob.

In summary, the problem asks for the probability that Z broke the dish given that a dish was broken. Using conditional probability, we find that P(Z|Br)= 0.05625. However, this answer differs from the solution given in the book (0.57), suggesting a possible mistake. Further calculations may be needed to confirm the correct answer.
  • #1
Bacle
662
1
Hi, everyone : I have the following problem:

We have 3 dishwashers X,Y,Z, with the conditions:



1) X washes 40% of dishes, and breaks
1% of the dishes s/he washes.

2)Y washes 30% of the dishes, breaks 1%

3)Z washes 30% of the dishes and breaks 3%.

Question: If a dish is broken: what is the probability that Z broke
the dish.?.

My work:
Events:
E1)Br means "Broke the dish",
E2)X (equiv. Y,Z) means "X ( Equiv. Y,Z) washed the dish.".
E3) (Br|X) means event of dish breaking when X is washing.

Notation:
P(A|B) is conditional probability of B, given A. '/\' means
intersection.


We have : P(Br|X)= 0.01 , P(Br|Y)=0.01 and P(Br|Z)=0.03

P(X)=0.4 , P(Y)=0.3 , P(Z)=0.3


We want to find P(Z|Br), which is equal to P(Z/\Br)/P(Br) , by def. of conditional
probability.


1) First, we find P(Br)=P( (Br/\X)\/(Br/\Y)\/(Br/\Z) )=

P(Br|X)P(X)+P(Br|Y)P(Y)+ p(Br|Z)P(Z)= 0.16

(side question: how do we know that any assignment of probabilities here will

give us P(Br)< =1 ? )


2) P(Z/\Br) =P(Z)P(Br|Z) = (0.3)(0.03)=0.009


3) Using 1,2 above, we get P(Br|Z)= 0.009/0.16= 9/160= 0.05625


But the book has 0.57 as a solution. Could the book have made a mistake.?
Would anyone please check.?

Thanks.
 
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  • #2
At first glance (without going through the math) you have dropped a power of 10 somewhere...

you may want to redo your calculations
 
  • #3
Thanks, Noobix.
You mean because Z should be much more likely to have broken the plate than either
X or Y (which would not be the case if P(Z|Br) was 0.057.)?
 

1. What is the difference between basic probabilities and conditional probabilities?

Basic probabilities refer to the likelihood of a single event occurring, while conditional probabilities refer to the likelihood of an event occurring given that another event has already occurred.

2. How do you calculate basic probabilities?

Basic probabilities can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

3. What is the formula for conditional probabilities?

The formula for conditional probabilities is P(A|B) = P(A and B) / P(B), where P(A|B) represents the probability of event A occurring given that event B has already occurred.

4. Can basic probabilities be greater than 1?

No, basic probabilities cannot be greater than 1. They represent a proportion or percentage and therefore must be between 0 and 1.

5. How are conditional probabilities used in real life?

Conditional probabilities are used in real life to make predictions and decisions based on existing information. For example, weather forecasts use conditional probabilities to predict the likelihood of rain given certain conditions such as temperature and humidity.

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