Conditional Probability Question

In summary, the question posed is what is the probability of getting at least one bad apple from tree A in a package of 3 apples, given that tree A produces 0.7 of the farm's apples and 0.15 of its apples are bad, while tree B produces 0.3 of the farm's apples and 0.05 of its apples are bad. Using a multinomial distribution, the probability is calculated to be 0.863.
  • #1
randomcat
7
0
Suppose that there are 2 apple trees. Tree A and tree B.

A produces 0.7 of the farm's apples. And B produces 0.3.

Out of the apples that tree A produces, 0.15 are bad. For B, 0.05 are bad.

One package of goodies contains 3 apples.

Given this information, what is the P(Tree A| at least one bad apple in the package)

Okay, so here's what I tried.

Probability of at least 1 bad apple in the package from tree A is:
x=0.7*(1-(0.85)^3)
Probability of at least 1 bad apple in the package from tree B is:
y=0.3*(1-(0.95)^3)

Then P(Tree A| at least one bad apple in the package) = x/(x+y) = 0.863

Does this seem right? Any help would be greatly appreciated.
 
Physics news on Phys.org
  • #2
An exact, but kludgy approach would be using a multinomial (4) distribution.
A = prob from tree A and good = .595 (no. = j)
a = prob from tree A and bad = .105 (no. = k)
B = prob from tree B and good = .285 (no. = m)
b = prob from tree B and bad = .015 (no. = n)

term = {3!/(j!k!m!n!)}AjakBmbn where j,k,m,n ≥ 0 and j+k+m+n = 3

Desired probability is fraction.
Numerator = sum of terms with k > 0 (contains a bad apple from A).
Denominator = sum of terms with k > 0 or n > 0 (contains a bad apple).
 

What is conditional probability?

Conditional probability is a mathematical concept that measures the likelihood of an event occurring given that another event has already occurred. It is expressed as the probability of A given B, or P(A|B).

How is conditional probability calculated?

Conditional probability is calculated by dividing the probability of the joint occurrence of two events (A and B) by the probability of the occurrence of the first event (B). This can be represented as P(A|B) = P(A and B) / P(B).

What is the difference between conditional probability and regular probability?

The main difference between conditional probability and regular probability is that conditional probability takes into account a specific condition or event that has already occurred, while regular probability does not consider any conditions or events.

How can conditional probability be used in real life?

Conditional probability is used in various fields, such as finance, insurance, and medicine, to make predictions and decisions based on existing data and conditions. For example, in medicine, conditional probability can be used to predict the likelihood of a patient developing a certain disease based on their medical history.

What is the importance of conditional probability in statistics?

Conditional probability is an important concept in statistics as it allows us to better understand the relationship between two events and make more accurate predictions based on available data. It is also used in many statistical models and techniques, such as regression analysis and Bayesian inference.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
10
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
771
  • Set Theory, Logic, Probability, Statistics
2
Replies
36
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
954
  • Set Theory, Logic, Probability, Statistics
Replies
6
Views
906
  • Set Theory, Logic, Probability, Statistics
Replies
12
Views
956
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
788
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
5
Views
1K
  • Set Theory, Logic, Probability, Statistics
2
Replies
41
Views
3K
Back
Top