Question

5. A game is played by rolling a die. If the die is a one or a two, a player draws a marble out of bag A. If the player rolls any other number, the player draws a marble out of bag B. Let A be the event the player draws a marble from bag A. Bag A has 8 marbles in it, 2 of which are black and 6 of which are red. Bag B has 12 marbles in it, 8 of which are black and 4 of which are red. After drawing a marble, the marble is put back in the bag. Let B be the event the marble is black. Draw a tree diagram to find the probability that on a player's turn, he or she draws a black marble.

Answer 1

Answer: Let's break down the probabilities step by step:

The player rolls a die:

If it's a 1 or 2 (probability = 2/6 = 1/3), they draw from bag A.

If it's any other number (probability = 4/6 = 2/3), they draw from bag B.

Probabilities for drawing a black marble from each bag:

Bag A: P(B|A) = Probability of drawing a black marble from bag A = 2 black / 8 total = 2/8 = 1/4

Bag B: P(B|B') = Probability of drawing a black marble from bag B = 8 black / 12 total = 8/12 = 2/3

Let's create the tree diagram:

1/3 (Drawing from Bag A)

/ \

A (1/4 Black) B' (2/3 Black)

(2/6 probability) (4/6 probability)

2/3 (Drawing from Bag B)

/ \

A' (1/4 Black) B (2/3 Black)

(4/6 probability) (4/6 probability)

Sure, I'll help you create a tree diagram to find the probability of drawing a black marble based on the given conditions.

Let's break down the probabilities step by step:

The player rolls a die:

If it's a 1 or 2 (probability = 2/6 = 1/3), they draw from bag A.

If it's any other number (probability = 4/6 = 2/3), they draw from bag B.

Probabilities for drawing a black marble from each bag:

Bag A: P(B|A) = Probability of drawing a black marble from bag A = 2 black / 8 total = 2/8 = 1/4

Bag B: P(B|B') = Probability of drawing a black marble from bag B = 8 black / 12 total = 8/12 = 2/3

Let's create the tree diagram:

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1/3 (Drawing from Bag A)

/ \

A (1/4 Black) B' (2/3 Black)

(2/6 probability) (4/6 probability)

2/3 (Drawing from Bag B)

/ \

A' (1/4 Black) B (2/3 Black)

(4/6 probability) (4/6 probability)

Now, let's find the overall probability of drawing a black marble:

P(B) = P(A and B) + P(B and B)

= P(A) * P(B|A) + P(B') * P(B|B')

= (1/3) * (1/4) + (2/3) * (2/3)

= 1/12 + 4/9

= 23/36

Therefore, the probability that on a player's turn, he or she draws a black marble is 23/36. Hope this helped!