Solve the given problem involving probability without replacement

  • #1
chwala
Gold Member
2,753
388
Homework Statement
This is apast paper question. Allow me to post it as it is.
Relevant Equations
Probability
1728080305462.png


Okay, i was able to solve it by trial and error, i am seeking for a more concrete approach. Can combination work here? or a more solid approach using sequences? or probability itself?

My trial and error,
##P_{green} = \dfrac{9}{12}×\dfrac{8}{11} ×\dfrac{7}{10}×\dfrac{6}{9}×\dfrac{3}{8} = \dfrac{21}{220}##

##n=5##.

and why 2 marks? or there is a shorter method.
 
  • Like
Likes Gavran
Physics news on Phys.org
  • #2
chwala said:
and why 2 marks?
And how should we know how many marks there are supposed to be? Is 2 not a perfect score for this question?
 
  • Like
Likes SammyS
  • #3
chwala said:
a more concrete approach
Excel:
1728096424538.png


The calculated value in C6 equals the target value in D6 when n=5.
 

Attachments

  • 1728096320696.png
    1728096320696.png
    2.9 KB · Views: 15
  • Like
Likes chwala
  • #4
Hill said:
Excel:
View attachment 351877

The calculated value in C6 equals the target value in D6 when n=5.
ok @Hill, I can see that you're still using the same approach as I am, but you're doing it differently. For instance, following your method,

##P(2) = \dfrac{9}{12}×\dfrac{3}{11}=0.204525##

and so on.

Cheers man!
 
  • #5
chwala said:
ok @Hill, I can see that you're still using the same approach as I am, but you're doing it differently. For instance, following your method,

##P(2) = \dfrac{9}{12}×\dfrac{3}{11}=0.204525##

and so on.

Cheers man!
These formulas are behind my numbers:
1728108388293.png


I.e., ##P(n)=\frac {3}{13-n}+\frac{10-n}{13-n}*P(n-1)##

In my approach, ##P(2) = 0.454545##, but rather ##P(2)-P(1)=0.204545##.
 
Last edited:
  • Like
Likes chwala
  • #6
## \begin{align}
P(n)&=\frac{9!}{(10-n)!}\cdot \frac{(12-n)!}{12!}\cdot3\nonumber\\
&=\frac{(12-n) \cdot (11-n)}{12\cdot 11\cdot 10}\cdot 3\nonumber\\
&=\frac{n^2-23n+132}{440}\nonumber
\end{align} ##

where ## P(n) ## is the probability that Marie picks a green ball on pick ## n ##.
 
  • Informative
Likes chwala
  • #7
I forgot to say that in my posts 3 and 5, ##P(n)## is the probability of picking a green ball on picks ##1## - ##n##. Thus, ##P(n)-P(n-1)## is the probability of picking a green ball on pick ##n##.
 
  • Like
Likes chwala and Gavran
  • #8
chwala said:
Homework Statement: This is apast paper question. Allow me to post it as it is.
Relevant Equations: Probability

View attachment 351870

Okay, i was able to solve it by trial and error, i am seeking for a more concrete approach. Can combination work here? or a more solid approach using sequences? or probability itself?

My trial and error,
##P_{green} = \dfrac{9}{12}×\dfrac{8}{11} ×\dfrac{7}{10}×\dfrac{6}{9}×\dfrac{3}{8} = \dfrac{21}{220}##

##n=5##.

and why 2 marks? or there is a shorter method.
I would do it this way.
 
  • Like
Likes chwala
  • #9
We can work backwards from [itex]\frac{21}{220}[/itex] until we get something we can recognise as [itex]P(n)[/itex] for some [itex]n[/itex]. We have a factor of 7 and a factor of 3 in the numerator, so we need at least factors of 9 and 8, which we get by multiplying by 9/9 = 1 and 8/8 = 1. We don't hae a factor of 12 in the denominator, but we can get that by mupltiying by 6/6 = 1 and combining the 6 with a factor of 2. Hence
[tex]\begin{split}
\frac{21}{220} &= \frac{7 \times 3}{2 \times 10 \times 11} \\
&= \frac{9}{9} \times \frac88 \times \frac{6}{6} \times \frac{7 \times 3}{2 \times 10 \times 11} \\
&= \frac{9}{6 \times 2} \times \frac{8}{11} \times \frac{7}{10} \times \frac{6}{9} \times \frac{3}{8} \\
&= P(5).\end{split}[/tex]
 
Last edited:
  • Informative
Likes chwala

FAQ: Solve the given problem involving probability without replacement

What does "without replacement" mean in probability?

"Without replacement" means that once an item is selected from a set, it is not returned to the set for further selection. This affects the total number of items available for subsequent selections, which in turn impacts the probabilities of future events.

How do I calculate the probability of drawing a specific item from a set without replacement?

To calculate the probability of drawing a specific item without replacement, divide the number of favorable outcomes (the specific item you want) by the total number of items in the set. After each draw, adjust the total number of items accordingly for subsequent draws, as the item is not replaced.

Can you provide an example of a probability problem without replacement?

Sure! If you have a bag containing 5 red balls and 3 blue balls, and you want to find the probability of drawing a red ball first and then a blue ball without replacement, you would calculate it as follows: The probability of drawing a red ball first is 5/8. After drawing one red ball, there are now 4 red balls and 3 blue balls left, making the total 7 balls. The probability of then drawing a blue ball is 3/7. Thus, the combined probability is (5/8) * (3/7) = 15/56.

How does the probability change with each draw without replacement?

The probability changes with each draw because the total number of items decreases. After each selection, the number of favorable outcomes and the total number of items are both updated, which alters the probabilities for subsequent draws. This means that the probabilities are dependent on the outcomes of previous draws.

What are common mistakes when solving probability problems without replacement?

Common mistakes include failing to adjust the total number of items after each draw, incorrectly calculating the number of favorable outcomes, and confusing the concept of "with replacement" versus "without replacement." It's important to keep track of how many items remain and ensure that the calculations reflect the reduced sample space after each selection.

Similar threads

Replies
1
Views
904
Replies
6
Views
1K
Replies
3
Views
5K
2
Replies
60
Views
9K
Replies
1
Views
1K
2
Replies
61
Views
9K
3
Replies
93
Views
13K
2
Replies
43
Views
11K
Back
Top