Probability of X: Geo Distribution with 2/3

In summary, we discussed a scenario where a fair cubical die with two yellow faces and four blue faces is rolled repeatedly until a yellow face appears. We found the probability of the blue face appearing exactly three times to be 8/81 and determined that the random variable X represents the number of times the blue face appears while the random variable Y represents the number of rolls. We calculated the probability distribution for X and found the expected value to be 130/81. We also found the probability of the die being rolled four times to be 8/27 and the probability of X and Y being equal to be 16/81. Finally, we considered a scenario where the game is repeated 200 times and found the probability of getting exactly
  • #1
songoku
2,266
319
Homework Statement
A fair cubical die has two yellow faces and four blue faces. In one game, the die is rolled repeatedly until yellow face appears uppermost. The maximum number of throw is 4. The random variable X represents the number of times blue face appears uppermost and random variable Y represents the number of times the die is rolled.
a. Find P(X = 3)
b. Find the probability distribution of X
c. Find E(X)
d. Find P(Y = 4)
e. P(Y = X)
f. If the game is repeated 200 times, write down (do not evaluate) an expression for the probability obtaining X = 3 exactly 25 times.
g. Use appropriate approximation, calculate the probability of question (f)
Relevant Equations
Not sure
a. P(X = 3) = (4/6) x (4/6) x (4/6) x (2/6) = 8/81

b. X ~ Geo (2/3)
Is this correct?

Thanks
 
Physics news on Phys.org
  • #2
songoku said:
Homework Statement:: A fair cubical die has two yellow faces and four blue faces. In one game, the die is rolled repeatedly until yellow face appears uppermost. The maximum number of throw is 4. The random variable X represents the number of times blue face appears uppermost and random variable Y represents the number of times the die is rolled.
a. Find P(X = 3)
b. Find the probability distribution of X
c. Find E(X)
d. Find P(Y = 4)
e. P(Y = X)
f. If the game is repeated 200 times, write down (do not evaluate) an expression for the probability obtaining X = 3 exactly 25 times.
g. Use appropriate approximation, calculate the probability of question (f)
Homework Equations:: Not sure

a. P(X = 3) = (4/6) x (4/6) x (4/6) x (2/6) = 8/81

b. X ~ Geo (2/3)
Is this correct?

Thanks

Yes, this is a set of Bernoulli trials with characteristic probability ##\frac 2 3##:

https://en.wikipedia.org/wiki/Geometric_distribution

Note the difference and similarity between the distributions of ##X## and ##Y##.

Sorry, I just saw that there is a maximum number of throws of ##4##. It's a truncated geometric distribution.
 
Last edited:
  • Like
Likes songoku
  • #3
Ok let me continue:

c. E(X) = q/p = 1/2

d. P( Y = 4) = (2/3)4 + (2/3)3 x (1/3) = 8/27

e. P (Y = X) = P(X = 4) = (2/3)4 = 16/81

f. Let random variable A represents number of games where X = 3
A ~ Binomial (n, p) so A ~ Binomial (200 , 8/81)
P (X = 25) = 200C25 (8/81)25 (73/81)75

Are all correct up until this? Thanks
 
  • #4
songoku said:
Ok let me continue:

c. E(X) = q/p = 1/2

d. P( Y = 4) = (2/3)4 + (2/3)3 x (1/3) = 8/27

e. P (Y = X) = P(X = 4) = (2/3)4 = 16/81

f. Let random variable A represents number of games where X = 3
A ~ Binomial (n, p) so A ~ Binomial (200 , 8/81)
P (X = 25) = 200C25 (8/81)25 (73/81)75

Are all correct up until this? Thanks

Note that there is a limit on the number of throws. I missed that. You need to calculate ##E(X)## for this.

When they ask for the distribution, I think they want the values for each outcome. There's a limit of four throws.
 
  • Like
Likes songoku
  • #5
PeroK said:
Note that there is a limit on the number of throws. I missed that. You need to calculate ##E(X)## for this.

When they ask for the distribution, I think they want the values for each outcome. There's a limit of four throws.
Not sure I understand what you mean but I did (c) using another way. I draw probability distribution table for X and I get this:
P(X = 0) = 1/3
P(X = 1) = 2/9
P(X = 2) = 4/27
P(X = 3) = 8/81
P(X = 4) = 16/81

Then E(X) = 0 . (1/3) + (1) (2/9) + (2) (4/27) + (3) (8/81) + (4) (16/81) = 130/81

Is this what you mean? I think this is the correct way to do (c). Thanks
 
  • Like
Likes PeroK
  • #6
songoku said:
Not sure I understand what you mean but I did (c) using another way. I draw probability distribution table for X and I get this:
P(X = 0) = 1/3
P(X = 1) = 2/9
P(X = 2) = 4/27
P(X = 3) = 8/81
P(X = 4) = 16/81

Then E(X) = 0 . (1/3) + (1) (2/9) + (2) (4/27) + (3) (8/81) + (4) (16/81) = 130/81

Is this what you mean? I think this is the correct way to do (c). Thanks

Yes, that all looks good.
 
  • Like
Likes songoku
  • #7
PeroK said:
Yes, that all looks good.
How about (d), (e), and (f)?
 
  • #8
songoku said:
How about (d), (e), and (f)?

Yes, they look correct.
 
  • Like
Likes songoku
  • #9
I think I can do (g)

Thank you very much perok
 
  • #10
I tell you, the probability of death is 100% :nb)
 

1. What does "Geo Distribution" mean in the context of probability?

Geo Distribution refers to a type of probability distribution that is used to model events or outcomes that occur in a spatial or geographical context. It takes into account the distance or location of the event or outcome from a specific reference point.

2. How is the probability of X calculated in a Geo Distribution with 2/3?

The probability of X in a Geo Distribution with 2/3 is calculated by taking into account the distance or location of the event or outcome from a specific reference point, as well as the probability of the event occurring at that distance. This is then multiplied by the overall probability of the event occurring, which is 2/3 in this case.

3. What does the term "2/3" represent in this probability distribution?

The term "2/3" represents the overall probability of the event occurring in the Geo Distribution. It indicates that out of all possible outcomes, the event will occur 2 times out of 3, or with a probability of 2/3.

4. How is a Geo Distribution with 2/3 different from other probability distributions?

A Geo Distribution with 2/3 is different from other probability distributions in that it takes into account the spatial or geographical context of events or outcomes. This can be useful in situations where the location of an event can affect its likelihood of occurring.

5. Can a Geo Distribution with 2/3 be used to predict future outcomes?

Yes, a Geo Distribution with 2/3 can be used to predict future outcomes as it takes into account the probability of events occurring at different distances from a reference point. However, it is important to note that it is not a perfect predictor and there may be other factors that can influence the outcome of an event.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
6
Views
544
  • Precalculus Mathematics Homework Help
Replies
11
Views
703
  • Precalculus Mathematics Homework Help
Replies
14
Views
2K
  • Precalculus Mathematics Homework Help
Replies
9
Views
921
  • Precalculus Mathematics Homework Help
Replies
2
Views
750
  • Precalculus Mathematics Homework Help
Replies
7
Views
1K
  • Precalculus Mathematics Homework Help
Replies
8
Views
513
  • Precalculus Mathematics Homework Help
Replies
2
Views
465
  • Precalculus Mathematics Homework Help
Replies
3
Views
1K
  • Precalculus Mathematics Homework Help
Replies
2
Views
1K
Back
Top