Question

1. suppose caylin has selected the other bag, the one with four white marbles and one red marble. what is the probability that caylin draws a red marble?

1. what is the probability that caylin has selected the bag with one white marble and four red marbles, then drawn a red marble?

1. the event that did occur is the observation that caylin draws a red marble. what is the probability that caylin draws a red marble from either bag?

1. based on this observation, what is the probability that the bag she selected is the one that contains one white marble and four red marbles? interpret this probability value, and explain why this probability is greater than \\(\frac{1}{2}\\).

Explanation:

Problem 3

Step1: Determine total and red marbles

The bag has 4 white and 1 red marble, so total marbles \( n = 4 + 1 = 5 \), red marbles \( r = 1 \).

Step2: Calculate probability

Probability \( P(\text{red}) = \frac{\text{Number of red marbles}}{\text{Total number of marbles}} = \frac{1}{5} \).

Step1: Probability of selecting the bag

Assume equal chance to select either bag, so \( P(\text{select bag with 1 white, 4 red}) = \frac{1}{2} \).

Step2: Probability of drawing red from that bag

In that bag, total marbles \( 1 + 4 = 5 \), red marbles \( 4 \), so \( P(\text{red}|\text{selected bag}) = \frac{4}{5} \).

Step3: Multiply the probabilities (multiplication rule for independent events)

\( P = \frac{1}{2} \times \frac{4}{5} = \frac{4}{10} = \frac{2}{5} \).

Step1: Define the two bags

Bag 1: 1 white, 4 red (total 5, red 4). Bag 2: 4 white, 1 red (total 5, red 1). Probability of selecting each bag is \( \frac{1}{2} \).

Step2: Calculate probability for each bag and sum

For Bag 1: \( P(\text{select Bag 1}) \times P(\text{red}|Bag 1) = \frac{1}{2} \times \frac{4}{5} = \frac{4}{10} \).
For Bag 2: \( P(\text{select Bag 2}) \times P(\text{red}|Bag 2) = \frac{1}{2} \times \frac{1}{5} = \frac{1}{10} \).
Total probability \( P = \frac{4}{10} + \frac{1}{10} = \frac{5}{10} = \frac{1}{2} \).

Answer:

\(\frac{1}{5}\)

Problem 4