Question

1. below are 2 boxes with black and white marbles. a box is selected and then a marble is drawn from the selected box. draw a tree diagram to and indicate on the diagram where you would find the probability of selecting black marble given it came from box 2. write the conditional probability equation and solve.

box 1: contains 2 white marbles (○) and 1 black marble (●)
box 2: contains 1 white marble (○) and 3 black marbles (●)
(handwritten notes: some markings near the boxes)

Explanation:

Step1: Define Events

Let \( B_1 \) be the event of selecting Box 1, \( B_2 \) be the event of selecting Box 2, \( B \) be the event of drawing a black marble.
We know \( P(B_1) = P(B_2) = \frac{1}{2} \) (since there are 2 boxes, equally likely to be selected).
In Box 1: 1 black, 2 white marbles. So \( P(B|B_1) = \frac{1}{3} \).
In Box 2: 3 black, 1 white marbles. So \( P(B|B_2) = \frac{3}{4} \).

Step2: Calculate \( P(B) \) (Law of Total Probability)

$$
LATEXBLOCK0
$$

Step3: Calculate \( P(B_2 \cap B) \)

$$ P(B_2 \cap B) = P(B|B_2)P(B_2) = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8} $$

Step4: Apply Conditional Probability Formula

The formula for conditional probability is \( P(B_2|B) = \frac{P(B_2 \cap B)}{P(B)} \)
Substitute the values:

$$
LATEXBLOCK1
$$

Answer:

The conditional probability that a black marble came from Box 2 is \(\frac{9}{13}\)