How to calculate some probabilities?

In summary, the conversation discussed calculating probabilities for different outcomes in a group of 10 "experiments" with 5 possible options. The probabilities for each option were given and it was explained that the order of appearance does not matter. To calculate the probability of a specific outcome, the probabilities of each option are multiplied together. It was also mentioned that the number of ways to distribute the letters in a given outcome must be taken into account when calculating the overall probability.
  • #1
Mastermind
5
0
I need help calculating some probabilities.

Here is my case:

There are 5 possibilities for an "experiment" : Right, Centre and Left, Up and down.

Chance of R = Chance of L = 26%
Chance of C = 38%
Chance of U = 8%
Chance of D = 2%

What is the possibility of each distribution in a group of 10 "experiments". The order of appearance doesn't matter.

i.e what's the probability of R=2, C = 4,L=4,U=1,D=1 (R+C+L+U+D=10), etc?

If it is too difficult/timeconsuming please explain how am I to calculate probability of each distribution when only R,C,L are possible. Group must be 10!
 
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  • #2
Mastermind said:
i.e what's the probability of R=2, C = 4,L=4,U=1,D=1 (R+C+L+U+D=10), etc?

2+4+4+1+1=12... :smile:

Lets call the probabalility for R pR, the probabalility for C pC etc...

The probabalility for a certain outcome, let's say RRCCLLLUUD is just pR*pR*pC*pC*pL*pL*pL*pU*pU*pD = 2,196E-8

If the order doesn't matter, so you just want to know what the chance is of getting 2 outcomes R, C and U, 3 times L and one D and you don't care in what order. You have to multiply the above calculated chance by the number of ways you can get this result. Because, and this is important, the chance on every such outcome is the same. This is, the chance to have RRCCLLLUUD as an outcome is exactly the same as the chance to get CRUULDLRCL...

But how many ways are there? RRCCLLLUUD, RCRCLLLUUD, LRRCCULLUUD, ... well a lot... If all 10 letters were different, it would be easy, it would be just 10!=10*9*8*7*6*5*4*3*2*1. But they are not the same and you have to account for that. If they are all different but two (for example the numer of ways to distribute the letters AABCDEFGHI where the order does not matter) there are 10!/2 ways to do it, because the sequences with the two A's interchanged are the same sequence. The number of ways to distribute AAABCDEFGH where order does not matter is 10!/6 or 10!/3! because there are 3! ways to interchange the A's

In general: The number of ways to distribute a sequence of 10 letters, where there are n letters who are the same and m other letters who are the same and o other letters who are the same (e.g. AAABBCCCDE here m=3, n=2 and o=2) is 10!/(m!*n!*o!*...).

So the number of ways to distribute RRCCLLLUUD where the order does not matter is 10!/(2!*2!*3!*2!*1!)= 48. So the chance for a sequence with 2 times R, 2 times C 3 times L, 2 times U and one time D is 48*2,196E-8=1,0541E-6

Good luck with calculating the rest!
 
  • #3


Calculating probabilities can seem intimidating at first, but it is actually quite simple once you understand the basic principles. In your case, you have 5 possible outcomes for each experiment, with different chances of occurring. To calculate the probability of a specific distribution in a group of 10 experiments, you will need to use the formula for calculating the probability of independent events, which is P(A and B) = P(A) x P(B).

Let's take your example of R=2, C=4, L=4, U=1, D=1. This means that out of the 10 experiments, R occurred twice, C occurred four times, L occurred four times, U occurred once, and D occurred once. To calculate the probability of this specific distribution, we need to multiply the individual probabilities for each outcome. So for R=2, the probability would be (0.26)^2 = 0.0676. Similarly, for C=4, the probability would be (0.38)^4 = 0.0276. For L=4, the probability would be (0.26)^4 = 0.0038. And for U=1 and D=1, the probabilities would be 0.08 and 0.02 respectively. To get the overall probability for this distribution, we multiply all of these individual probabilities together: 0.0676 x 0.0276 x 0.0038 x 0.08 x 0.02 = 0.0000004888. This is the probability of getting the specific distribution R=2, C=4, L=4, U=1, D=1 in a group of 10 experiments.

If you want to calculate the probability for a different distribution, you can follow the same process. Just remember to use the corresponding probabilities for each outcome. And if you only want to consider R, C, and L as possible outcomes, you can simply ignore the probabilities for U and D when calculating the overall probability.

I hope this explanation helps you understand how to calculate probabilities for different distributions. Remember to always use the formula P(A and B) = P(A) x P(B) and to adjust the probabilities accordingly based on the number of times each outcome occurs. Good luck!
 

1. How do I calculate the probability of a single event occurring?

To calculate the probability of a single event occurring, divide the number of favorable outcomes by the total number of possible outcomes. This will give you a decimal value between 0 and 1, which can be converted to a percentage by multiplying by 100.

2. How do I calculate the probability of multiple independent events occurring?

To calculate the probability of multiple independent events occurring, multiply the individual probabilities of each event. This is known as the "AND" rule, where the probability of both events occurring is equal to the product of their individual probabilities.

3. How do I calculate the probability of at least one of multiple events occurring?

To calculate the probability of at least one of multiple events occurring, use the "OR" rule. This means adding together the individual probabilities of each event, then subtracting the probability of both events occurring at the same time (which was already included in the addition). This gives you the total probability of at least one of the events occurring.

4. How do I calculate the probability of dependent events occurring?

To calculate the probability of dependent events occurring, you will need to use the "AND" rule. However, the probability of the second event will be affected by the outcome of the first event. This means that the probability of both events occurring is equal to the probability of the first event multiplied by the probability of the second event given that the first event has occurred.

5. How do I calculate the probability of mutually exclusive events occurring?

To calculate the probability of mutually exclusive events occurring, use the "OR" rule. This means adding together the individual probabilities of each event, but since the events are mutually exclusive (cannot occur at the same time), you do not need to subtract any overlap. The sum of the individual probabilities will give you the total probability of at least one of the events occurring.

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