Question

1. a bag of jellybeans has 8 yellow, 6 red, 4 black, and 7 green jellybeans. if you eat two jellybeans one at a time, what is the probability of eating a red jellybean followed by a green one?

a. \\(\frac{13}{25}\\)

b. 0.0672

c. \\(\frac{7}{100}\\)

d. 0.007

Explanation:

🆕 New Concept Discovered: Dependent Events Probability
Finding the chance of consecutive events when the first outcome changes the next.

Step 1: Identify the total number of jellybeans

First, find the total number of jellybeans in the bag by adding all the colors together:

$$ \text{Total} = 8 \text{ (yellow)} + 6 \text{ (red)} + 4 \text{ (black)} + 7 \text{ (green)} = 25 \text{ jellybeans} $$

Step 2: Find the probability of eating a red jellybean first

The probability of choosing a red jellybean first is the number of red jellybeans divided by the total number of jellybeans:

$$ P(\text{Red}_1) = \frac{6}{25} $$

Step 3: Find the probability of eating a green jellybean second

Since you eat the first jellybean, it is not replaced. This means both the number of green jellybeans and the total number of jellybeans change for the second draw:

• The number of green jellybeans remains \( 7 \).
• The total number of jellybeans remaining in the bag is now \( 25 - 1 = 24 \).

The probability of choosing a green jellybean second, given that a red one was eaten first, is:

$$ P(\text{Green}_2 \mid \text{Red}_1) = \frac{7}{24} $$

Step 4: Calculate the combined probability

To find the probability of both events happening in sequence, multiply the two individual probabilities:

$$ P(\text{Red}_1 \text{ and then } \text{Green}_2) = \frac{6}{25} \times \frac{7}{24} $$

Simplify the calculation by cross-canceling \( 6 \) and \( 24 \):

$$ \frac{6}{25} \times \frac{7}{24} = \frac{1}{25} \times \frac{7}{4} = \frac{7}{100} $$

Converting this fraction to a decimal:

$$ \frac{7}{100} = 0.07 $$

Comparing this with the given options:

• A. \( \frac{13}{25} \)
• B. \( 0.0672 \)
• C. \( \frac{7}{100} \)
• D. \( 0.007 \)

The correct option is C.

Answer:

C. \( \frac{7}{100} \)