# Finding the probability of given model

I'm trying to model a system given as a project, any help or advice is useful for now.

Here our assumptions for a part of it:

• If I study for the finals and do assignments, I have $80\%$ chance to get a scholarship
• If I neither do assignments nor study, my chance reduces to $20\%$
• If I either do the assignments or study for the final, I have $60\%$ chance to get a scholarship.
• The chance that I will do assignments or study are both $50\%$.

So, given that I got the scholarship what is the probability that I have did the assignments?

• I think there's something wrong in the question, as it contains the answer. Your 4th bullet point says the probability you do the assignments is 50%. If you assume that, then nothing else changes it. – Peter Flom May 19 '13 at 15:31
• @Peter Flom, can you just explain why is it so ? – berkay May 19 '13 at 15:35
• There's nothing to explain. You assumed it. – Peter Flom May 19 '13 at 16:14

I would suggest drawing a tree diagram to visualize this problem (A = Assignments, F = Finals, S = Scholarship): The red probabilities at the end of the branches are the products of probabilities leading to these branches. For example for the leftmost branch we have $0.5\times 0.5\times 0.8 = 0.2$. From this it is easy to calculate the probability: There are four possibilities of getting a scholarship, but only two of those include you having done your assignments (marked with boxes). So the probability of having done the assignments given that you received a scholarship is:

$$P(A|S) = \frac{0.2+0.15}{0.2+0.15+0.15+0.05}=\frac{0.35}{0.55}\approx 0.636$$

It is the sum of probabilities of those two branches of the tree where you got scholarship and did the assignments divided by the total sum of probabilities of getting the scholarship.

• i'm clear now, thanks. which tool have you used for the visualization? – berkay May 19 '13 at 16:50
• This is a beautiful answer but not to the question, as posed. Given the 4th bullet point, no further modeling or analysis is needed. The probability is 0.5. If you delete the 4th bullet point, then this answer is correct. – Peter Flom May 19 '13 at 16:56
• @berkay You're welcome. I drew the tree in PowerPoint, but I'm sure that there are better tools for this around. I think Bayes theorem lies at the heart of the question. See this PDF for a nice tutorial. – COOLSerdash May 19 '13 at 21:19

The conditional probability of having done assignments given that you got the scholarship is $35/55\approx0.636$. The probability of having done assignments is $0.5$, as stated by the fourth bullet point.

You have 4 equally likely scenarios a priori:

1. study + assignments $\rightarrow\, 80\%$ probability of getting the scholarship
2. study + no assignments $\rightarrow\, 60\%$ probability of getting the scholarship
3. no study + assignments $\rightarrow\, 60\%$ probability of getting the scholarship
4. do nothing $\rightarrow\, 20\%$ probability of getting the scholarship

The conditional probability is $P(A|S)=\dfrac{0.8+0.6}{0.8+2\centerdot0.6+0.2}=\dfrac{35}{55}\approx0.636$

• thanks but how do you calculate equation --> 35/55=0.636 – berkay May 19 '13 at 16:23
• I updated my answer to include the line of thought. – Marc Claesen May 19 '13 at 16:39
• i accepted the other answer for visual representation but thanks for your help. all i can do is +1 :) – berkay May 19 '13 at 16:46