Questions: A bag contains 3 pink gumballs, 4 red gumballs, and 2 green gumballs. Find the probability that at least one red gumball is drawn if there are 5 drawings that occur and the gumballs are replaced after each draw. Round your answer to two decimal places.

Solution Steps

We have a bag containing \(3\) pink gumballs, \(4\) red gumballs, and \(2\) green gumballs. The total number of gumballs is:

\[ 3 + 4 + 2 = 9 \]

We want to find the probability of drawing at least one red gumball in \(5\) drawings, with replacement after each draw.

The probability of drawing a red gumball in one draw, denoted as \(p\), is given by:

\[ p = \frac{\text{Number of red gumballs}}{\text{Total number of gumballs}} = \frac{4}{9} \]

The probability of not drawing a red gumball, denoted as \(q\), is:

\[ q = 1 - p = 1 - \frac{4}{9} = \frac{5}{9} \]

To find the probability of drawing at least one red gumball in \(5\) draws, we first calculate the probability of drawing no red gumballs in all \(5\) draws. This is represented as \(P(X = 0)\) in a binomial distribution:

\[ P(X = 0) = \binom{n}{0} \cdot p^0 \cdot q^n = 1 \cdot 1 \cdot \left(\frac{5}{9}\right)^5 \]

Calculating \(P(X = 0)\):

\[ P(X = 0) = \left(\frac{5}{9}\right)^5 \approx 0.0529 \]

The probability of drawing at least one red gumball is given by:

\[ P(X \geq 1) = 1 - P(X = 0) = 1 - 0.0529 \approx 0.9471 \]

Final Answer