Drawing Red Balls

Solution & Explanation

To determine the probability of drawing at least 2 red balls when drawing 3 balls from a bag containing 3 red balls and 7 blue balls, we can consider the following scenarios:

1. Drawing exactly 2 red balls and 1 blue ball
2. Drawing all 3 red balls

Total Number of Possible Outcomes

The total number of ways to choose 3 balls out of 10 is given by the combination formula:

$C(10, 3) = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120$

Scenario 1: Drawing Exactly 2 Red Balls and 1 Blue Ball

• First, calculate the number of ways to choose 2 red balls from the 3 available red balls:

  $C(3, 2) = \frac{3 \times 2}{2 \times 1} = 3$

• Next, calculate the number of ways to choose 1 blue ball from the 7 available blue balls:

  $C(7, 1) = 7$

• The total number of favorable outcomes for this scenario is:

  $C(3, 2) \times C(7, 1) = 3 \times 7 = 21$

Scenario 2: Drawing All 3 Red Balls

• Calculate the number of ways to choose all 3 red balls from the 3 available red balls:

  $C(3, 3) = 1$

• The total number of favorable outcomes for this scenario is:

  $1$

Probability Calculation

• The probability of drawing exactly 2 red balls and 1 blue ball is:

  $\frac{21}{120}$

• The probability of drawing all 3 red balls is:

  $\frac{1}{120}$

• Therefore, the probability of drawing at least 2 red balls (either exactly 2 or all 3) is:

  $\frac{21}{120} + \frac{1}{120} = \frac{22}{120}$

• Simplifying the fraction:

  $\frac{22}{120} = \frac{11}{60}$

• Thus, the probability of drawing at least 2 red balls is:

  $\frac{11}{60} \approx 0.183$

This probability indicates that there is approximately an 18.3% chance of drawing at least 2 red balls when drawing 3 balls from the bag.