• MHB
• Monoxdifly
In summary, the value of 2m + 3n is 70. This is determined by the given information that there are m white balls and n red balls in a bag, with mn = 200 and more white balls than red balls. Additionally, the probability of taking two different colored balls when two are randomly chosen at once is \frac{40}{87}. Solving for m and n, it is found that m = 20 and n = 10, resulting in 2m + 3n = 70.
Monoxdifly
MHB
In a bag there are m white balls and n red balls with mn = 200 and there are more white balls than red balls. If two balls are taken randomly at once and the probability of taking two different colored balls is $$\displaystyle \frac{40}{87}$$ then the value of 2m + 3n is ...
A. 30
B. 45
C. 50
D. 70
E. 80

Okay, so the possibility of m and n are like this:
m = 200 and n =1
m = 100 and n = 2
m = 50 and n = 4
m = 40 and n = 5
m = 25 and n = 8
m = 20 and n = 10

Do I need to count their probability one by one then adding them up to make $$\displaystyle \frac{40}{87}$$? Or am I not supposed to do that?

P(two different colors) =

$\dfrac{m}{m+n} \cdot \dfrac{n}{m+n-1} + \dfrac{n}{m+n} \cdot \dfrac{m}{m+n-1} = \dfrac{40}{87}$

$\dfrac{2mn}{(m+n)(m+n-1)} = \dfrac{40}{87}$

$\dfrac{400}{k(k-1)} = \dfrac{40}{87}$, where $k = m+n$

$k = m+n = 30 \text{ and } mn = 200 \implies m = 20 \text{ and } n = 10$

Took me a while to understand that k = 30 comes from the factorization of $$\displaystyle k^2-k-870=0$$, but thank you. Now I understand. :)

## 1. What is probability with factors?

Probability with factors is a mathematical concept that involves determining the likelihood of a certain event occurring, taking into account various factors that may influence the outcome. These factors can include variables, conditions, or other elements that may affect the probability of the event.

## 2. How is probability with factors calculated?

The calculation of probability with factors involves multiplying the probability of each individual factor together. This is known as the multiplication rule of probability. For example, if the probability of event A occurring is 0.5 and the probability of event B occurring is 0.3, then the probability of both events occurring together is 0.5 x 0.3 = 0.15.

## 3. What is the difference between independent and dependent factors in probability?

Independent factors in probability refer to events that do not affect each other and can occur simultaneously. In other words, the outcome of one event does not influence the outcome of the other. On the other hand, dependent factors in probability are events that are connected and the outcome of one event does affect the outcome of the other.

## 4. How can probability with factors be used in real-life situations?

Probability with factors can be used in a variety of real-life situations, such as predicting the outcome of a sports game, determining the likelihood of a medical diagnosis, or calculating the chances of winning a lottery. It can also be used in decision-making processes, such as determining the best course of action in a business scenario.

## 5. Can probability with factors be applied to non-numerical situations?

Yes, probability with factors can be applied to non-numerical situations, such as predicting the outcome of a legal case or determining the likelihood of success for a new product launch. In these cases, the factors may not have numerical values, but they can still be taken into account to calculate the overall probability of a certain event occurring.

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