Questions: A gumball machine contains 7 red gumballs and 4 white gumballs. Two gumballs are purchased, one after the other, without replacement. Find the probability that at least one gumball is white. Express your answer as a decimal, rounded to the nearest hundredth. Type your answer.-

Solution Steps

We have a gumball machine containing a total of \( N = 11 \) gumballs, consisting of \( K = 4 \) white gumballs and \( 7 \) red gumballs. We want to find the probability of drawing at least one white gumball when two gumballs are drawn without replacement.

To find the probability of drawing at least one white gumball, we first calculate the probability of drawing 0 white gumballs. This can be expressed using the hypergeometric distribution formula:

\[ P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}} \]

For our case, where \( k = 0 \) (0 white gumballs drawn), \( n = 2 \) (total gumballs drawn), we have:

\[ P(X = 0) = \frac{\binom{4}{0} \binom{7}{2}}{\binom{11}{2}} = \frac{1 \cdot 21}{55} = 0.3818 \]

The probability of drawing at least one white gumball is the complement of drawing 0 white gumballs:

\[ P(X \geq 1) = 1 - P(X = 0) = 1 - 0.3818 = 0.6182 \]

Rounding this to two decimal places gives us:

\[ P(X \geq 1) \approx 0.62 \]

Final Answer