Calculating Probability of Extracting 5 Pairs of Different Colored Balls

In summary, we have a box with 5 white and 5 black balls. Two balls are extracted at a time, and the random variable X represents the number of extractions with the same colour. We can have a range of 0 to 5 for X, with 0 representing all extractions resulting in different coloured pairs and 5 representing all extractions resulting in the same coloured pairs. The probability of X=0 is 8/63 and the probability of X=2 is 40/63, with X=4 having a probability of 15/63. To calculate the number of all possible events, we can use the formula nCr = n! / r!(n-r)!, where n is the total
  • #1
etf
179
2
Hi!
Here is my task:
There are 5 white and 5 black balls in box. Two balls at the time are extracted from box. Let random variable X be "number of extractions of balls with same colour. Find probability of random variable.

x can go from 0 (5 times we extract two by two balls and they are with different colour) to 5 (5 times we extract two by two balls and they are with same colour). I started with P(x=0) and I stucked there. How to calculate number of all possible events when extracting two by two balls at the time and number of all possible events of interest (we extract 5 pairs of balls and they are with different colours)?
Few events of interest woul be for example:
black ball white ball, black ball white ball, black ball white ball, black ball white ball,black ball white ball or
white ball black ball, white ball black ball, white ball black ball, white ball black ball, white ball black ball or
white ball black ball, black ball white ball, black ball white ball, black ball white ball, white ball black ball
etc.
 
Last edited:
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  • #2
Result is:
$$X: \begin{pmatrix}
0 & 2 & 4 \\
8/63 & 40/63 &15/63
\end{pmatrix} $$
 
  • #3
etf said:
Hi!
Here is my task:
There are 5 white and 5 black balls in box. Two balls at the time are extracted from box. Let random variable X be "number of extractions of balls with same colour. Find probability of random variable.

x can go from 0 (5 times we extract two by two balls and they are with different colour) to 5 (5 times we extract two by two balls and they are with same colour). I started with P(x=0) and I stucked there. How to calculate number of all possible events when extracting two by two balls at the time and number of all possible events of interest (we extract 5 pairs of balls and they are with different colours)?
Few events of interest woul be for example:
black ball white ball, black ball white ball, black ball white ball, black ball white ball,black ball white ball or
white ball black ball, white ball black ball, white ball black ball, white ball black ball, white ball black ball or
white ball black ball, black ball white ball, black ball white ball, black ball white ball, white ball black ball
etc.

Show some work. For example, how would you compute the probability that all pairs have different colours?
 

1. How do you calculate the probability of extracting 5 pairs of different colored balls?

To calculate the probability of extracting 5 pairs of different colored balls, you need to first determine the total number of possible combinations. This can be done by finding the total number of balls and the total number of colors. Then, you can use the formula (n!/(r!(n-r)!))^5, where n is the total number of balls and r is the total number of colors. This will give you the total number of possible combinations. To find the probability, divide the number of desired outcomes (in this case, 5 pairs of different colored balls) by the total number of possible combinations.

2. What factors affect the probability of extracting 5 pairs of different colored balls?

The two main factors that affect the probability of extracting 5 pairs of different colored balls are the total number of balls and the total number of colors. The more balls and colors there are, the lower the probability becomes. Other factors that can affect the probability include the method of extraction (e.g. with or without replacement) and any biases or patterns in the ball selection process.

3. Can you provide an example of calculating the probability of extracting 5 pairs of different colored balls?

Sure, let's say there are 10 red balls, 8 blue balls, and 6 green balls in a bag. The total number of balls is n = 10 + 8 + 6 = 24 and the total number of colors is r = 3. Plugging these values into the formula, we get (24!/(3!(24-3)!))^5 = (2024/6)^5 = 337^5. So, the total number of possible combinations is 337^5. If we want to find the probability of extracting 5 pairs of different colored balls, we divide the desired outcome (5 pairs) by the total number of combinations, which is (5/337^5) or approximately 3.85 x 10^-9.

4. What are some real-world applications of calculating the probability of extracting 5 pairs of different colored balls?

Calculating the probability of extracting 5 pairs of different colored balls can be useful in various fields, including gambling, statistics, and quality control. In gambling, it can help players make more informed decisions based on the likelihood of certain outcomes. In statistics, it can be used to determine the likelihood of certain events occurring in a sample population. In quality control, it can be used to assess the reliability and consistency of a manufacturing process.

5. What are some limitations of calculating the probability of extracting 5 pairs of different colored balls?

There are a few limitations to consider when calculating the probability of extracting 5 pairs of different colored balls. Firstly, the formula assumes that all balls have an equal chance of being selected, which may not always be the case. Additionally, the method of extraction (with or without replacement) can affect the probability. Also, the formula does not account for any external factors that may influence the selection process. Lastly, the probability is based on theoretical calculations and may not always reflect the actual outcomes in real-world scenarios.

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