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A bag contains 5 red, 3 green, 4 blue, and 8 yellow marbles. Find the probability of randomly selecting a green marble, and then a yellow marble if the first marble is replaced.

$1$ . A bag contains $5$ red, $3$ green, $4$ blue, and $8$ yellow marbles. Find the probability of randomly selecting a green marble, and then a yellow marble if the first marble is replaced.

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Probability of independent and dependent events

Full solution

Q.

1

. A bag contains

5

red,

3

green,

4

blue, and

8

yellow marbles. Find the probability of randomly selecting a green marble, and then a yellow marble if the first marble is replaced.

Calculate Total Marbles: Total number of marbles = $5$ red + $3$ green + $4$ blue + $8$ yellow = $20$ marbles.
Calculate Probability of Green: Probability of picking a green marble = Number of green marbles / Total number of marbles = $\frac{3}{20}$ .
Marble Replacement: Since the marble is replaced, the total number of marbles remains the same for the second pick.
Calculate Probability of Yellow: Probability of picking a yellow marble after replacing the first = Number of yellow marbles / Total number of marbles = $\frac{8}{20}.$
Multiply Probabilities: To find the combined probability of two independent events, multiply the probabilities of each event.
Calculate Combined Probability: Combined probability = Probability of green first $\times$ Probability of yellow second = $(\frac{3}{20}) \times (\frac{8}{20})$ .
Simplify Fraction: Combined probability = $(\frac{3}{20}) \times (\frac{8}{20}) = \frac{24}{400}$ .
Simplify Fraction: Combined probability = $(\frac{3}{20}) \times (\frac{8}{20}) = \frac{24}{400}$ . Simplify the fraction $\frac{24}{400}$ to its lowest terms.
Simplify Fraction: Combined probability = $(\frac{3}{20}) \times (\frac{8}{20}) = \frac{24}{400}$ . Simplify the fraction $\frac{24}{400}$ to its lowest terms. $24$ and $400$ are both divisible by $8$ , so $\frac{24}{400} = \frac{3}{50}$ .

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