Probability of Increasing Order in Multiple Dice Rolls

  • Thread starter Mandelbroth
  • Start date
  • Tags
    Probability
In summary: That may be easier.In summary, the question is to find the probability that A\leq B\leq C\leq D when rolling a fair 9-sided die four times and assigning values to each roll. One approach is to find the probability that A\leq B and then extend it to greater numbers of rolls. Another approach is to break it down according to the number of rolls that are the same.
  • #1
Mandelbroth
611
24

Homework Statement


A friend proposed this question to me. It looks like a homework problem, though, so I'll put it here to avoid any conflict with forum rules.

Suppose you roll a fair 9 sided die (with sides 1 through 9) 4 times. Let A be the value of the first roll, B be the value of the second roll, C be the value of the third roll, and D be the value of the fourth roll.

Find the probability that [itex]A\leq B\leq C\leq D[/itex]

The Attempt at a Solution


I figure that the probability that [itex]A\leq B[/itex] is [itex]\frac{9+36}{81}=\frac{45}{81}[/itex] because there are 9 ways that they can be equal and half of the remaining 72 outcomes have A less than B. However, I'm having difficulty extending this to greater numbers of rolls.

Can someone please help guide me toward my next step? Thank you, in advance.
 
Physics news on Phys.org
  • #2
Mandelbroth said:

Homework Statement


A friend proposed this question to me. It looks like a homework problem, though, so I'll put it here to avoid any conflict with forum rules.

Suppose you roll a fair 9 sided die (with sides 1 through 9) 4 times. Let A be the value of the first roll, B be the value of the second roll, C be the value of the third roll, and D be the value of the fourth roll.

Find the probability that [itex]A\leq B\leq C\leq D[/itex]

The Attempt at a Solution


I figure that the probability that [itex]A\leq B[/itex] is [itex]\frac{9+36}{81}=\frac{45}{81}[/itex] because there are 9 ways that they can be equal and half of the remaining 72 outcomes have A less than B. However, I'm having difficulty extending this to greater numbers of rolls.

Can someone please help guide me toward my next step? Thank you, in advance.

Letting ##E = \{A \leq B \leq C \leq D\}## we have
[tex] P(E) = \sum_{a=1}^9 P(A=a) P(E | A=a),[/tex]
and for a = 1,2,3, ..., 9 we have
[tex] P(E|A=a) = P(a \leq B \leq C \leq D ).[/tex]

Let ##F_a = \{ a \leq B \leq C \leq D\}##. We have
[tex] P(F_a) = \sum_{b=a}^9 P(B=b) P(F_a | B = b) ,[/tex]
and for b = a, ..., 9 we have
[tex] P(F_a | B = b) = P(b \leq C \leq D).[/tex]
Keep going like that, or else try to find a nice formula for ##Q(b) \equiv P(b \leq C \leq D).## Then ##P(E) = \sum_{a=1}^9 \sum_{b=a}^9 P(A=a)P(B=b) Q(b). ##
 
  • #3
Another approach is to break it down according to numbers of rolls the same. There are 9*8*7*6 (ordered) rolls in which they're all different, and 1/24th of these will be in the desired order. 4C2*9*8*7 with one pair the same, of which 2/24 are in the right order. Then there's the two pairs case, the 3 of a kind, and four of a kind.
 

What is probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 represents impossibility and 1 represents certainty.

How is probability calculated?

Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. This is known as the probability formula: P(A) = number of favorable outcomes / total number of possible outcomes.

What is the difference between theoretical and experimental probability?

Theoretical probability is based on mathematical calculations and assumes that all outcomes are equally likely. Experimental probability is based on actual observations and may vary from the theoretical probability due to chance or other factors.

What is the difference between independent and dependent events?

Independent events are events that do not affect each other, meaning the outcome of one event does not influence the outcome of the other. Dependent events are events that do affect each other, meaning the outcome of one event is dependent on the outcome of the other.

How can probability be used in real life?

Probability can be used in many real-life situations, such as predicting the likelihood of a certain outcome in a game or sports event, estimating the chances of winning a lottery or raffle, and making decisions based on risk assessment. It is also used in various fields of science, including medicine, economics, and psychology.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
2
Views
1K
  • Precalculus Mathematics Homework Help
Replies
11
Views
2K
  • Precalculus Mathematics Homework Help
Replies
9
Views
3K
  • Precalculus Mathematics Homework Help
2
Replies
53
Views
5K
  • Precalculus Mathematics Homework Help
Replies
3
Views
1K
  • Precalculus Mathematics Homework Help
Replies
9
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
6
Views
1K
  • Precalculus Mathematics Homework Help
Replies
13
Views
6K
  • Precalculus Mathematics Homework Help
Replies
1
Views
1K
  • Precalculus Mathematics Homework Help
Replies
2
Views
2K
Back
Top