use the information to answer the question. mr. williams has a bag filled with 3 red pencils and 12 blue…

use the information to answer the question. mr. williams has a bag filled with 3 red pencils and 12 blue pencils that are all the same size. for a prize, a student randomly chooses two pencils from the bag. what is the probability that the first pencil is red and the second pencil is blue? justify your answer.

use the information to answer the question. mr. williams has a bag filled with 3 red pencils and 12 blue pencils that are all the same size. for a prize, a student randomly chooses two pencils from the bag. what is the probability that the first pencil is red and the second pencil is blue? justify your answer.

Answer

Explanation:

Step1: Find total number of pencils

First, calculate the total number of pencils. There are 3 red and 12 blue, so total is (3 + 12 = 15) pencils.

Step2: Probability first is red

The probability that the first pencil is red is the number of red pencils divided by total pencils. So (P(\text{first red})=\frac{3}{15}=\frac{1}{5}).

Step3: Probability second is blue (after first red)

After removing one red pencil, there are (15 - 1 = 14) pencils left, and 12 blue pencils. So (P(\text{second blue}|\text{first red})=\frac{12}{14}=\frac{6}{7}).

Step4: Multiply the probabilities

Since these are dependent events, multiply the two probabilities: (P=\frac{1}{5}\times\frac{6}{7}=\frac{6}{35}).

To justify: We use the multiplication rule for dependent events. The first event (choosing red) affects the second event (choosing blue) because the total number of pencils decreases by 1 after the first pick. So we calculate the probability of the first event, then the probability of the second event given the first occurred, and multiply them.

Answer:

The probability is (\frac{6}{35}). The justification is that we calculate the probability of choosing a red pencil first ((\frac{3}{15})) and then a blue pencil from the remaining 14 pencils ((\frac{12}{14})), then multiply these probabilities for dependent events, resulting in (\frac{3}{15}\times\frac{12}{14}=\frac{6}{35}).