Probability of Getting 3 A's & 2 B's in 5 Rounds

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In summary, the lucky machine has three results: A,B,C. The probability of getting A is 0.2, the probability of getting B is 0.3, and the probability of getting C is 0.5. You played five rounds and got three results: A,B,C.
  • #1
oriel1
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There is lucky machine with 3 results (letters):
A,B,C
P(A)=0.2 (Probabilty to get A)
P(B)=0.3 "" "" ""
P(C)=0.5
Each round you get just one letter.
You played 5 times.
What the probability that you got three times "A" and two times "B".

i thinked to do that:
\(\displaystyle 0.2^{3}\)\(\displaystyle \cdot*\)\(\displaystyle 0.3^{2}\)\(\displaystyle \frac{{5}!}{{2}!\left({3}\right)!}\)
\(\displaystyle \frac{{5}!}{{2}!\left({3}\right)!}\) is the number of options to get that result.
But it doesn't make any sense. because \(\displaystyle \frac{{5}!}{{2}!\left({3}\right)!}\)=10 and if the probabilty is higher then 0.1 the result will be >1 and it can't happen.
There is any good solution for this problem ? binomial distribution or something?
Thank you.
 
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  • #2
Hi oriel,

Your calculation looks good to me! You found the probability of one sequence of 3 A's and 2 B's, then multiplied by the number of sequences. Indeed if $(.2^3 \cdot .3^2)$ were greater than .1, then we would have a problem, but it isn't. If you multiply your answer out, we get a valid probability. This isn't proof that the answer is correct, but it isn't outside the range of a probability calculation.
 
  • #3
I was typing an answer but then I saw Jameson's post and he basically gave the answer. Your calculation is fine! Now, I want to add a comment to your final question: can we use a binomial distribution here? The answer is no. The binomial distribution can only be used in experiments where you have two possible outcomes (a failure and a success). Here you have three. However, you can use the multinomial distribution which is a generalization of the binomial distribution for $k$ different outcomes. If you're interested then I suggest you read the wikipedia page:
https://en.wikipedia.org/wiki/Multinomial_distribution

You'll notice that your calculation is exactly how the probability would be calculated if you had explicitly used the multinomial distribution.

Finally, the probability is indeed small ... do you find the result surprising?
 
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  • #4
Jameson said:
Hi oriel,

Your calculation looks good to me! You found the probability of one sequence of 3 A's and 2 B's, then multiplied by the number of sequences. Indeed if $(.2^3 \cdot .3^2)$ were greater than .1, then we would have a problem, but it isn't. If you multiply your answer out, we get a valid probability. This isn't proof that the answer is correct, but it isn't outside the range of a probability calculation.
Siron said:
I was typing an answer but then I saw Jameson's post and he basically gave the answer. Your calculation is fine! Now, I want to add a comment to your final question: can we use a binomial distribution here? The answer is no. The binomial distribution can only be used in experiments where you have two possible outcomes (a failure and a success). Here you have three. However, you can use the multinomial distribution which is a generalization of the binomial distribution for $k$ different outcomes. If you're interested then I suggest you read the wikipedia page:
https://en.wikipedia.org/wiki/Multinomial_distribution

You'll notice that your calculation is exactly how the probability would be calculated if you had explicitly used the multinomial distribution.

Finally, the probability is indeed small ... do you find the result surprising?
Thank you very much for that clear explanation. I appreciate that.
 

FAQ: Probability of Getting 3 A's & 2 B's in 5 Rounds

1. What is the probability of getting 3 A's and 2 B's in 5 rounds?

The probability of getting 3 A's and 2 B's in 5 rounds depends on the total number of possible outcomes and the likelihood of each outcome. In order to calculate the probability, you would need to know the number of A's and B's in the deck and the number of rounds being played. Using this information, you can use the binomial probability formula to determine the probability.

2. How is probability calculated in this scenario?

In order to calculate the probability of getting 3 A's and 2 B's in 5 rounds, you would use the binomial probability formula: P(x) = nCx * p^x * (1-p)^(n-x), where n is the total number of trials, x is the number of desired outcomes, and p is the probability of success for each trial. In this case, n = 5, x = 3, and p = (number of A's in deck / total number of cards).

3. Is it possible to increase the probability of getting 3 A's and 2 B's in 5 rounds?

It is possible to increase the probability of getting 3 A's and 2 B's in 5 rounds by adjusting the number of A's and B's in the deck and the number of rounds being played. However, the probability will always be dependent on the total number of possible outcomes and the likelihood of each outcome.

4. How does the number of rounds affect the probability?

The number of rounds being played will affect the probability as it increases the number of trials in the binomial probability formula. This means that the more rounds you play, the higher the probability of getting 3 A's and 2 B's in 5 rounds will be. However, the probability will still depend on the total number of possible outcomes and the likelihood of each outcome.

5. Can probability be used to guarantee a specific outcome of 3 A's and 2 B's in 5 rounds?

No, probability cannot be used to guarantee a specific outcome of 3 A's and 2 B's in 5 rounds. It can only determine the likelihood of this outcome occurring based on the given information and the laws of probability. There is always a chance for other outcomes to occur, even if they may be less likely.

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