Part A: We need to develop a probability model based on the given information (100 people bought gum balls, 45 red, 40 blue, 15 yellow). We also need to compare the probability of getting a red gum ball to the probability of getting a yellow gum ball. Part B: Given that the next 10 people got 7 yellow, 1 red, and 2 blue, we need to explain a possible reason for this outcome given the probabilities from the original 100 people.

1. Problem Description

Part A: We need to develop a probability model based on the given information (100 people bought gum balls, 45 red, 40 blue, 15 yellow). We also need to compare the probability of getting a red gum ball to the probability of getting a yellow gum ball.

Part B: Given that the next 10 people got 7 yellow, 1 red, and 2 blue, we need to explain a possible reason for this outcome given the probabilities from the original 100 people.

2. Solution Steps

Part A:

First, we need to determine the probabilities of getting each color of gum ball. The total number of people is

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0. The probability of getting a red gum ball, $P(Red)$, is the number of red gum balls divided by the total number of gum balls purchased:

$P(Red) = \frac{45}{100} = 0.45$

The probability of getting a blue gum ball, $P(Blue)$, is:

$P(Blue) = \frac{40}{100} = 0.40$

The probability of getting a yellow gum ball, $P(Yellow)$, is:

$P(Yellow) = \frac{15}{100} = 0.15$

To compare the probability of getting a red gum ball to the probability of getting a yellow gum ball, we can express the ratio of these probabilities:

$\frac{P(Red)}{P(Yellow)} = \frac{0.45}{0.15} = 3$

This means the probability of getting a red gum ball is 3 times higher than the probability of getting a yellow gum ball.

Part B:

The probabilities from the first 100 people are: $P(Red) = 0.45$, $P(Blue) = 0.40$, and $P(Yellow) = 0.15$.

The next 10 people got 7 yellow, 1 red, and 2 blue. This outcome deviates significantly from the initial probabilities. A possible reason for this is that the gum ball machine was not well-mixed, and the yellow gum balls were clustered in one section. Therefore, a small sample of 10 people may have gotten more yellow balls than expected based on the initial distribution. Another possible reason is that the supplier refilled the machine with a different proportion of colors compared to the initial fill.

3. Final Answer

Part A:

The probability model is: $P(Red) = 0.45$, $P(Blue) = 0.40$, $P(Yellow) = 0.15$. The probability of getting a red gum ball is 3 times the probability of getting a yellow gum ball.

Part B:

A possible reason for this outcome is that the gum balls in the machine were not well mixed, leading to a disproportionate number of yellow gum balls being dispensed to the next few people. Another possible reason is that the gum ball machine was refilled with a batch that has a much larger proportion of yellow gum balls.