There are 50 small balls in a tub. Some balls are white and some are orange. Without being able to see into the tub, each student in a class of 25 is allowed to pick a ball out of the tub at random. The color of the ball is recorded and the ball is put back into the tub. At the end, 7 orange balls and 18 white balls were picked. What is the best estimate you can give for the number of orange balls and the number of white balls in the tub? Describe how to calculate this best estimate, and explain your method of calculation makes sense in a way that a seventh grader might understand. Is your best estimate necessarily accurate? Why or why not?
Based on the sampling of 25, we can estimate that the chance of drawing an orange ball is 7/25.
Let's say we did a sampling of 50. This tells us that, since 50 is 2 x 25, we could estimate that the chance of drawing an orange ball is then 14/50.
Some students might note that we are returning the balls each time, so it doesn't really represent the actual 50 balls that are in the bag. So we can also think about it this way: there appear to be 18 white balls for every 7 orange balls. 18+7 =25. So we could double these values and estimate that there are 36 white balls for every 14 orange balls. 36 +14=50.
Lastly, we can note that a best estimate isn't necessarily correct because we need many more trials for accuracy. While not probable, it is possible that there is 1 orange ball and 49 white balls in the tub!
To find the best estimate for the number of orange and white balls in the tub, we can use a method called proportion estimation.
Step 1: Calculate the proportion of orange and white balls that were picked.
- Proportion of orange balls picked: Number of orange balls picked / Number of total balls picked = 7 / 25 = 0.28
- Proportion of white balls picked: Number of white balls picked / Number of total balls picked = 18 / 25 = 0.72
Step 2: Determine the number of balls in the tub.
Since each student picked a ball at random, we can assume that the proportion of balls they picked represents the proportion of balls in the tub.
- Number of orange balls in tub = Proportion of orange balls picked * Total number of balls in tub = 0.28 * 50 = 14
- Number of white balls in tub = Proportion of white balls picked * Total number of balls in tub = 0.72 * 50 = 36
So, the best estimate is that there are about 14 orange balls and 36 white balls in the tub.
However, it's important to note that this estimate may not be completely accurate. It's based on the assumption that the sample of 25 balls picked by the students is representative of the entire tub. If the sample size were larger, the estimate would likely be more accurate.
The estimate also assumes that each ball has an equal chance of being picked, which may not be the case if the balls have different sizes or weights that affect their probability of being chosen.
Additionally, the estimate does not consider any external factors that could have influenced the picking, such as the placement of balls or any biases in picking.
Therefore, while the best estimate provides a reasonable approximation, it's not necessarily accurate due to these limitations.
To estimate the number of orange and white balls in the tub, we will use a method called "proportional reasoning." Proportional reasoning involves using the information available to make a reasonable guess based on what we know.
Here's how we can calculate the best estimate:
1. We know that there are 25 students in the class, and each student picked a ball. So, the total number of balls picked is 25.
2. Out of those 25 balls, 7 are orange and 18 are white.
3. To estimate the total number of orange balls in the tub, we can set up a proportion. Since 7 orange balls were picked out of 25 total balls, we can say that the ratio of orange balls picked to total balls picked is 7:25.
4. To find the estimated total number of orange balls in the tub, we use the same ratio and apply it to the total number of balls in the tub. Let's call the total number of orange balls "x." So, we have the proportion: 7/25 = x/50.
5. To solve for "x," we can cross-multiply and solve the equation. Thus, 7 * 50 = 25 * x. Simplifying this, we get 350 = 25x. Dividing both sides by 25, we find that x = 14.
Therefore, our best estimate for the number of orange balls in the tub is 14.
To estimate the number of white balls, we can subtract the estimated number of orange balls (14) from the total number of balls (50): 50 - 14 = 36.
Hence, our best estimate for the number of white balls in the tub is 36.
It's important to note that the best estimate we obtained is not necessarily accurate. Since the students were picking the balls randomly, the sample of 25 picks might not accurately represent the entire tub of balls. It is possible that if we repeated this experiment many times, we would get slightly different results each time. However, by using proportional reasoning and making the assumption that the proportions remain the same, we can make a reasonable guess or estimate.