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$A bag contains 5 red marbles, 4 green marbles, 6 white marbles, and 5 black marbles. What is the probability of randomly drawing a red marble and then a green marble if the first marbled is replaced? Write your answer as a fraction in simplest form.$

A bag contains $5$ red marbles, $4$ green marbles, $6$ white marbles, and $5$ black marbles. What is the probability of randomly drawing a red marble and then a green marble if the first marbled is replaced? Write your answer as a fraction in simplest form.

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Probability of independent and dependent events

Full solution

Q. A bag contains

5

red marbles,

4

green marbles,

6

white marbles, and

5

black marbles. What is the probability of randomly drawing a red marble and then a green marble if the first marbled is replaced? Write your answer as a fraction in simplest form.

Calculate Total Marbles: Calculate the total number of marbles in the bag. $\newline$ The bag contains $5$ red marbles, $4$ green marbles, $6$ white marbles, and $5$ black marbles. To find the total, we add these numbers together. $\newline$ $5$ (red) $+$ $4$ (green) $+$ $6$ (white) $+$ $5$ (black) $4$ $1$ $4$ $2$ marbles.
Probability of Red Marble: Determine the probability of drawing a red marble first. $\newline$ Since there are $5$ red marbles out of $20$ total marbles, the probability of drawing a red marble is: $\newline$ Probability of red marble = Number of red marbles / Total number of marbles = $\frac{5}{20}$ . $\newline$ We can simplify this fraction to $\frac{1}{4}$ .
Marble Replacement: Since the first marble is replaced, the total number of marbles in the bag remains the same for the second draw. This means that the probability of drawing a green marble after replacing the first marble is still based on the original count of marbles.
Probability of Green Marble: Determine the probability of drawing a green marble second. Since there are $4$ green marbles out of the original $20$ marbles, the probability of drawing a green marble is: Probability of green marble $= \frac{\text{Number of green marbles}}{\text{Total number of marbles}} = \frac{4}{20}$ . We can simplify this fraction to $\frac{1}{5}$ .
Combined Probability: Calculate the combined probability of drawing a red marble first and then a green marble after replacing the first marble. $\newline$ To find the combined probability of two independent events, we multiply their probabilities together. $\newline$ Combined probability = Probability of red marble $\times$ Probability of green marble = $\frac{1}{4}$ $\times$ $\frac{1}{5}$ = $\frac{1}{20}$ .

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