# What Are Common Mistakes in Solving Conditional Probability Problems?

• Woolyabyss
In summary, the conversation discusses questions related to a teacher giving two tests to a class and the percentage of students who passed both tests or only one test. It also includes a question about the probability of a defective item being produced by different machines in a factory. The calculated probabilities are 0.9 for those who passed both tests and 0.02 for the probability of a defective item being produced by machine Q.
Woolyabyss

## Homework Statement

Question 1:
A teacher gave his class two tests where every student passed at least one test. 72% of the class passed passed both tests and 80% of the class passed the second test.
(i)what percentage of those who passed the second test also passed the first?
(ii) what percentage of the class passed the first but failed the second test?

Question 2
A factory has three machines P,Q and R, producing large numbers of a certain item Of the total production, 40% is produced on P, 50% on Q and 10% on R. The records show that 1% of the items produced on P are defective, 2% of items produced on Q are defective and 6% of items of items produced on R are defective The occurrence of a defective item is independent of each machine and all other items.
(i)calculate the probability the item chosen is defective.
(ii) Given that the item chosen is defective, find the probability that it was produced on machine
Q.

## Homework Equations

P(A|B) = P(AnB)/P(B)

## The Attempt at a Solution

1. (i)

P(F|S) = .72/.8 = .9

(ii)

Since all students pass at least one test.

P(F U S) = 1

1= P(F) + P(S) - P(FnS)... P(F) = 0.92 all people who passed the first test

P(F) - P(FnS) = 0.2 This is the number who only passed the first test.

According to the answers in my book I got the first part right but the second part wrong.

Question 2
(i)
PD = .4(.01)
QD = .5(.02)
RD = .1(.06)

(ii)

P(Q|D) = .5(.02)/.5 = .02

According to my book both of these answers are wrong.Any help would be appreciated.

Woolyabyss said:
According to the answers in my book I got the first part right but the second part wrong.
I'd say the book is wrong.
Looks right to me.
(ii)

P(Q|D) = .5(.02)/.5 = .02
Think again about what you are dividing by here.

haruspex said:
I'd say the book is wrong.Looks right to me.
Think again about what you are dividing by here.

Would it be P(Q|D) = .5(.02)/.02 = .5 ??

Woolyabyss said:
Would it be P(Q|D) = .5(.02)/.02 = .5 ??

Yes. Do you see how that follows from the equation?

haruspex said:
Yes. Do you see how that follows from the equation?

Ya, I didnt divide by the probability of q being defective the first time. Thanks

## 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 P(A|B), which reads as "the probability of event A given event B".

## How is conditional probability calculated?

Conditional probability is calculated by dividing the probability of the joint occurrence of both events (A and B) by the probability of the occurrence of the first event (B). In formula form, it looks like this: P(A|B) = P(A∩B) / P(B).

## What is the difference between conditional and unconditional probability?

Conditional probability takes into account the occurrence of a specific event, while unconditional probability does not. In other words, conditional probability considers the probability of an event given that another event has already occurred, while unconditional probability looks at the probability of an event occurring without any prior knowledge or condition.

## How is conditional probability used in real life?

Conditional probability is used in a variety of fields, including science, economics, and statistics. It can help predict the likelihood of certain outcomes based on specific conditions, such as the probability of a disease given certain risk factors or the probability of a stock market crash given certain economic indicators.

## What are some common misconceptions about conditional probability?

Some common misconceptions about conditional probability include assuming that events that occur together are always causally related, assuming that past events have no influence on future events, and assuming that conditional probability is the same as causation. It's important to remember that correlation does not equal causation and that conditional probability is just one tool for understanding relationships between events.

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