cynara has a bag of red, blue, and black pens. she randomly selects a pen, records the color, and puts it…

cynara has a bag of red, blue, and black pens. she randomly selects a pen, records the color, and puts it back in the bag. she does this a total of 30 times and selects 6 red pens, 10 blue pens, and 14 black pens. based on this information, what is the probability that cynara will select a blue pen, return it to the bag, and then select a red pen?

cynara has a bag of red, blue, and black pens. she randomly selects a pen, records the color, and puts it back in the bag. she does this a total of 30 times and selects 6 red pens, 10 blue pens, and 14 black pens. based on this information, what is the probability that cynara will select a blue pen, return it to the bag, and then select a red pen?

Answer

Explanation:

Step1: Calculate probability of blue pen

The probability of an event is the number of favorable outcomes divided by the total number of outcomes. The total number of pen - selection trials is 30. The number of times a blue pen was selected is 10. So the probability of selecting a blue pen, $P(B)=\frac{10}{30}=\frac{1}{3}$.

Step2: Calculate probability of red pen

The number of times a red pen was selected is 6. So the probability of selecting a red pen, $P(R)=\frac{6}{30}=\frac{1}{5}$.

Step3: Calculate the combined probability

Since the pen is replaced after the first selection, the two events are independent. For independent events $A$ and $B$, the probability of both $A$ and $B$ occurring is $P(A)\times P(B)$. So the probability of selecting a blue pen and then a red pen is $P = P(B)\times P(R)$. Substitute $P(B)=\frac{1}{3}$ and $P(R)=\frac{1}{5}$ into the formula: $P=\frac{1}{3}\times\frac{1}{5}=\frac{1}{15}$.

Answer:

$\frac{1}{15}$