Question

question 13 points 3

the probability of yi choosing a white and an orange ball is \\(\frac{8}{45}\\). what is the probability that yi chose an orange ball second, given that he chose a white ball first? there are 4 white balls and total number of balls is 10.

\\(\bigcirc \frac{32}{45}\\)
\\(\bigcirc \frac{4}{9}\\)

Explanation:

Define the events and given values

Using the Conditional Probability knowledge point
Let \(W_1\) be the event that Yi chooses a white ball first.
Let \(O_2\) be the event that Yi chooses an orange ball second.
Let \(W\) and \(O\) represent choosing one white and one orange ball in any order.
We are given:

• Total number of balls: \(N = 10\)
• Number of white balls: \(N_W = 4\)
• Probability of choosing a white and an orange ball: \(P(W \text{ and } O) = \frac{8}{45}\)

Calculate the probability of choosing a white ball first

Using the Conditional Probability knowledge point

$$ P(W_1) = \frac{N_W}{N} = \frac{4}{10} = \frac{2}{5} $$

Relate the joint probability to the ordered outcomes

Using the Conditional Probability knowledge point
Since the selection is done without replacement, choosing a white and an orange ball can happen in two mutually exclusive orders:

1. White first, then orange: \(W_1 \cap O_2\)
2. Orange first, then white: \(O_1 \cap W_2\)

Let \(x\) be the number of orange balls.
The probability of choosing a white then an orange ball is:

$$ P(W_1 \cap O_2) = \frac{4}{10} \times \frac{x}{9} = \frac{4x}{90} $$

The probability of choosing an orange then a white ball is:

$$ P(O_1 \cap W_2) = \frac{x}{10} \times \frac{4}{9} = \frac{4x}{90} $$

Thus, the total probability of choosing one white and one orange ball is:

$$ P(W \text{ and } O) = P(W_1 \cap O_2) + P(O_1 \cap W_2) = \frac{4x}{90} + \frac{4x}{90} = \frac{8x}{90} = \frac{4x}{45} $$

Solve for the number of orange balls

Using the Conditional Probability knowledge point
We equate the calculated probability to the given probability:

$$ \frac{4x}{45} = \frac{8}{45} \implies 4x = 8 \implies x = 2 $$

So, there are \(2\) orange balls.

Calculate the conditional probability

Using the Conditional Probability knowledge point
We want to find the probability that Yi chose an orange ball second, given that he chose a white ball first:

$$ P(O_2 \mid W_1) = \frac{P(W_1 \cap O_2)}{P(W_1)} $$

Using the values calculated:

$$ P(W_1 \cap O_2) = \frac{4}{10} \times \frac{2}{9} = \frac{8}{90} = \frac{4}{45} $$

$$ P(O_2 \mid W_1) = \frac{\frac{4}{45}}{\frac{2}{5}} = \frac{4}{45} \times \frac{5}{2} = \frac{2}{9} $$

Alternatively, since a white ball is chosen first, there are now \(9\) balls remaining, of which \(2\) are orange:

$$ P(O_2 \mid W_1) = \frac{2}{9} $$

Answer:

• (A) \(\frac{32}{45}\)
• (B) \(\frac{2}{9}\) (Correct answer)
• (C) \(\frac{4}{9}\)