Conditional Probability Question

In summary, the problem involves two urns with white and yellow balls, and the task is to draw 3 balls without replacement from one of the urns based on a roll of an ordinary six-sided die. The probability of drawing exactly two white balls is calculated for both urns, and the overall probability is found by combining the probabilities for each urn based on the probability of rolling a 1 or a 2. Finally, the probability of rolling a 1 given that exactly two white balls were drawn is calculated.
  • #1
laz0r
17
0

Homework Statement


Suppose there are two urns that contain white and yellow balls. Urn 1 contains
10 white and 5 yellow balls, and Urn 2 contains 6 white and 12 yellow balls. You
are going to draw 3 balls without replacement from one of the urns. To decide
which urn to draw from, you will roll an ordinary six-sided (balanced) die. If the
die comes up with a 1, you will draw 3 balls from Urn 1, otherwise you will draw
3 balls from Urn 2.
(a) What is the probability you draw exactly two white balls?
(b) Given you draw exactly two white balls, what is the probability you rolled a
1?


Homework Equations


(10,2) for instance would read ten choose 2 in combinatorics

The Attempt at a Solution



Ok, so I'm not entirely sure how to do this, but this is my attempt..

Urn #1)
10 white balls
5 yellow

((10,2)(5,1))/(15,3) = P(choose exactly 2 white balls from urn #1) = P(#1) = 45/91

Urn #2)

6 white balls
12 yellow

((6,2)(12,1))/(18,3) = P(choose exactly 2 white balls from urn #2) = P(#2) = 15/68

-----

P(draw from urn 1) = 1/6 = P(#11)
P(draw from urn 2) = 5/6 = P(#22)

P(draw exactly 2 white balls) = P(#1)P(#11) + P(#2)P(#22) = (45/91)(1/6) + (15/68)(5/6) = 0.26624

PART B)

P(roll a 1 | draw 2 white balls) = P(roll a 1 and draw 2 white balls)/P(draw 2 white balls)

Am I on the right track here?
 
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  • #2
Wrong kind of question for this forum: PF is not in the business of stamp-approving homework. Wouldn't be good for our relations with the poor teachers of this world.

Show some self-assurance !

I have no clue what would be the right track, but I enjoyed following yours. Thanks for sharing this with us ! :approve:
 

Related to Conditional Probability Question

1. What is conditional probability?

Conditional probability is a mathematical concept that calculates the likelihood of an event occurring given that another event has already occurred. It takes into account the new information provided by the first event and adjusts the probability accordingly.

2. How is conditional probability calculated?

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

3. What is the difference between conditional probability and regular probability?

Regular probability calculates the likelihood of an event occurring without taking into account any additional information. Conditional probability, on the other hand, adjusts the probability based on new information provided by another event.

4. Can conditional probability be greater than regular probability?

Yes, conditional probability can be greater than regular probability because it takes into account additional information. This can change the likelihood of an event occurring.

5. What are some real-world examples of conditional probability?

One example of conditional probability is the likelihood of a person having a certain disease given that they test positive on a diagnostic test. Another example is the probability of a flight being delayed given that there is bad weather in the area.

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