Question

as you answer the questions below, consider how the probability of an event can be updated based on new observations.

brendan has two bags of marbles. the two bags look the same, and each contains five marbles. in one bag, there are one white marble and four red marbles. in the other bag, there are four white marbles and one red marble. caylin randomly selects one bag of marbles. she cannot see the marbles in the bag, and only knows the marbles she has drawn. she then randomly draws a marble from the bag she selected. she sees that it is a red marble. 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?

1. before drawing any marbles, explain why the probability that caylin selected the bag that contains one white marble and four red marbles is \\(\frac{1}{2}\\).

1. suppose caylin has selected the bag with one white marble and four red marbles. what is the probability that caylin draws a red marble? explain why this is a conditional probability.

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?

Explanation:

Question 1

Step1: Identify the total number of bags

There are 2 bags in total, and Caylin selects one bag randomly.

Step2: Determine the probability of selecting a specific bag

Since the selection is random and there are 2 equally - likely bags (the bag with 1 white and 4 red marbles and the other bag), the probability of selecting the bag with 1 white and 4 red marbles is the number of favorable outcomes (1 bag) divided by the total number of outcomes (2 bags). So the probability is $\frac{1}{2}$.

Step1: Identify the number of marbles in the bag

The bag with 1 white and 4 red marbles has a total of $1 + 4=5$ marbles.

Step2: Calculate the probability of drawing a red marble

The number of red marbles is 4. The probability of drawing a red marble from this bag is the number of red marbles divided by the total number of marbles in the bag. So the probability $P(\text{Red}|\text{Bag1})=\frac{4}{5}$. This is a conditional probability because we are finding the probability of drawing a red marble given that Caylin has selected a specific bag (the bag with 1 white and 4 red marbles). The "given that" (the selection of the specific bag) is the condition.

Step1: Identify the number of marbles in the bag

The bag with 4 white and 1 red marble has a total of $4 + 1 = 5$ marbles.

Step2: Calculate the probability of drawing a red marble

The number of red marbles is 1. The probability of drawing a red marble from this bag is the number of red marbles divided by the total number of marbles in the bag. So the probability $P(\text{Red}|\text{Bag2})=\frac{1}{5}$.

Answer:

The probability is $\frac{1}{2}$ because there are 2 equally - likely bags to choose from, and we are interested in 1 of them.

Question 2