Probability Theory: Simultaneous picks

In summary, the conversation is about a problem involving picking balls of the same color and using the geometric distribution. The person is struggling to understand the method of counting probabilities for successive and simultaneous picks. They are looking for assistance in understanding the concept. The solution involves using the same math for both cases, resulting in a chance of obtaining balls of the same color of (20*19+10*9)/(30*29).
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Homework Statement


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Hi all, I have an issue understanding the concepts pertaining to the following problem, assistance is greatly appreciated.
Screen Shot 2018-09-17 at 7.38.41 PM.png


I understand the "flow" of the problem; first find the probability of obtaining balls of the same colour, then use the geometric distribution. However, I don't understand why the method of counting probabilities doesn't change between the cases of successive picks, and simultaneous picks.

The way I always understood outcome-counting for successive picks (number of ways of picking 2 red balls w/o replacement for instance) was that the first pick had 20 available options, the second 19. Then divide by 2 to account for the fact that we don't bother with order, so
$$\frac{20\times 19}{2} = {20\choose2}$$
The case of simultaneous picks doesn't come as intuitively to me, I have tried to find ways to understand it (like "how many ways can i touch 20 red balls with my 2 hands") but every time it makes sense, it doesn't.

Could anyone assist? Many thanks!

Homework Equations

The Attempt at a Solution

 

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  • #2
"Simultaneous picks" doesn't change the math. Your chance of same color remains (20*19+10*9)/(30*29).
 

1. What is probability theory and how does it apply to simultaneous picks?

Probability theory is a branch of mathematics that deals with the study of random events. It is used to analyze and predict the likelihood of an event occurring. In the context of simultaneous picks, probability theory is used to calculate the chances of multiple outcomes happening at the same time.

2. How is the probability of simultaneous picks calculated?

The probability of simultaneous picks is calculated by multiplying the individual probabilities of each event. For example, if there is a 50% chance of picking a red ball and a 25% chance of picking a blue ball, the probability of picking both a red and blue ball at the same time would be 50% * 25% = 12.5%.

3. What is the difference between independent and dependent events in probability theory?

In probability theory, independent events are those that do not affect each other's probability of occurrence. On the other hand, dependent events are those that are affected by the outcome of a previous event. In simultaneous picks, the probability of each pick is affected by the previous picks, making them dependent events.

4. How can probability theory be applied in real-life situations?

Probability theory can be applied in various real-life situations, such as predicting the outcome of sports games, weather forecasting, and risk assessment in financial investments. It is also used in decision-making processes, such as in business and healthcare, to weigh the likelihood of different outcomes and make informed choices.

5. What are some limitations of using probability theory in simultaneous picks?

One limitation of using probability theory in simultaneous picks is that it assumes all events are independent, which may not always be the case in real-life situations. Additionally, the accuracy of the predicted probabilities depends on the accuracy of the data and assumptions used in the calculations. It is also important to note that probability theory cannot guarantee a specific outcome, but rather provides a likelihood of it occurring.

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