Question

A box contains 20 red balls, 25 white balls, and 55 blue balls. Suppose that 10 balls are selected at random one at a time with replacement; that is, each selected ball is replaced in the box before the next selection is made. Determine the probability that at least one color will be missing from the 10 selected balls.

Answer 1

The problem involves calculating the probability of selecting balls of at least two different colors from a box while considering the possibility of not selecting one of the colors. The solution requires the application of the binomial distribution formula and the inclusion-exclusion principle to account for the scenarios where one or more colors might be missing.

Step-by-step explanation:

The question involves determining the probability that at least one color will be missing when 10 balls are selected with replacement from a box containing 20 red balls, 25 white balls, and 55 blue balls. To solve this, we consider all possible outcomes and subtract the probability that all colors are selected at least once.

First, we calculate the probability of selecting only two colors in 10 draws. We have three scenarios for missing one color: (1) no red balls, (2) no white balls, (3) no blue balls. For each scenario, we use the binomial distribution because balls are drawn with replacement, which means the probability of drawing each color remains constant for each draw.

To calculate these probabilities we add the probabilities of the three scenarios and then subtract this sum from 1, which will give us the required probability. However, since these scenarios can overlap (we can miss two colors at the same time), we would need to apply the inclusion-exclusion principle to get the correct total probability.

Answer 2

0.164 = 16.4% probability that at least one color will be missing from the 10 selected balls.

Step-by-step explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

Determine the probability that at least one color will be missing from the 10 selected balls.

Red missing:

Each time, there are 55 + 25 = 80 non-red balls, out of 100. So, in each of the 10 trials, 80% = 0.8 probability of not picking a red ball. The probability that no red ball is picked is given by:

(0.8)^10 = 0.1074

White missing:

55 + 20 = 75 non-white balls, out of 100, in each trial. The probability that no white ball is picked is given by:

(0.75)^10 = 0.0563

Blue missing:

45 non-blue balls, out of 100. The probability that no blue ball is picked is given by:

(0.45)^10 = 0.0003

Total:

0.1074 + 0.0563 + 0.0003 = 0.164

0.164 = 16.4% probability that at least one color will be missing from the 10 selected balls.