Probability of Selecting at Least Four Boys

Oct 13, 2025 by wbrum.org 32 views

Probability of Selecting at Least Four Boys

In this problem, we are given that approximately 40% of the boys in Walter's school commute to school by bicycle. We need to find the probability that we must select at least four boys to find one boy who commutes by bicycle.

Defining the Problem

Let's assume that there are a total of 100 boys in Walter's school. We can use the concept of probability to solve this problem. The probability of selecting a boy who commutes by bicycle is 40%, which can be represented as 0.4.

Calculating the Probability

To find the probability that we must select at least four boys to find one boy who commutes by bicycle, we need to calculate the probability of the complementary event, which is the probability that we select three boys who do not commute by bicycle.

Complementary Event

The probability that a boy does not commute by bicycle is 1 - 0.4 = 0.6. Since we are selecting three boys, the probability that all three boys do not commute by bicycle is (0.6)^3 = 0.216.

Probability of Selecting at Least Four Boys

Now, we need to find the probability that we select at least four boys to find one boy who commutes by bicycle. This can be represented as 1 - P(3 boys do not commute by bicycle). Therefore, the probability is 1 - 0.216 = 0.784.

Interpretation of Results

The probability that we must select at least four boys to find one boy who commutes by bicycle is 0.784 or 78.4%. This means that there is a 78.4% chance that we will need to select at least four boys to find one boy who commutes by bicycle.

Conclusion

In conclusion, we have used the concept of probability to solve this problem. We have calculated the probability that we must select at least four boys to find one boy who commutes by bicycle and found that it is 0.784 or 78.4%.

Table of Randomly Generated Numbers

To answer this problem, we can use a table of randomly generated numbers. Let's assume that we have a table of 100 randomly generated numbers, where each number represents a boy in Walter's school. We can use this table to simulate the selection of boys and calculate the probability.

Boy #	Commutes by Bicycle?
1	Yes
2	No
3	No
4	Yes
5	No
6	Yes
7	No
8	No
9	Yes
10	No
...	...
100	Yes

Simulation

We can simulate the selection of boys by randomly selecting numbers from the table. For example, if we select the first 10 numbers, we get:

Boy #	Commutes by Bicycle?
1	Yes
2	No
3	No
4	Yes
5	No
6	Yes
7	No
8	No
9	Yes
10	No

In this simulation, we need to select 10 boys to find one boy who commutes by bicycle. We can repeat this simulation many times to get an estimate of the probability.

Estimating the Probability

After repeating the simulation many times, we can estimate the probability that we must select at least four boys to find one boy who commutes by bicycle. Let's assume that we repeat the simulation 1000 times and get the following results:

Number of Boys Selected	Frequency
1	0
2	0
3	216
4	784
5	0
...	...

From this table, we can see that the probability that we must select at least four boys to find one boy who commutes by bicycle is approximately 0.784 or 78.4%.

Conclusion

Q: What is the probability that we must select at least four boys to find one boy who commutes by bicycle?

A: The probability that we must select at least four boys to find one boy who commutes by bicycle is 0.784 or 78.4%.

Q: How did you calculate the probability?

A: We calculated the probability by using the concept of complementary events. We first found the probability that a boy does not commute by bicycle, which is 0.6. Then, we calculated the probability that all three boys do not commute by bicycle, which is (0.6)^3 = 0.216. Finally, we found the probability that we must select at least four boys to find one boy who commutes by bicycle by subtracting the probability of the complementary event from 1.

Q: What is the significance of the probability?

A: The probability that we must select at least four boys to find one boy who commutes by bicycle is significant because it gives us an idea of how many boys we need to select to find one who commutes by bicycle. In this case, we need to select at least four boys to find one who commutes by bicycle.

Q: Can you explain the concept of complementary events?

A: Yes, the concept of complementary events is a fundamental concept in probability theory. It states that the probability of an event occurring is equal to 1 minus the probability of the event not occurring. In this case, the event is selecting a boy who commutes by bicycle, and the complementary event is selecting a boy who does not commute by bicycle.

Q: How did you use the table of randomly generated numbers to simulate the selection of boys?

A: We used the table of randomly generated numbers to simulate the selection of boys by randomly selecting numbers from the table. For example, if we select the first 10 numbers, we get a list of boys who commute by bicycle or not. We can repeat this simulation many times to get an estimate of the probability that we must select at least four boys to find one boy who commutes by bicycle.

Q: What is the advantage of using a table of randomly generated numbers to simulate the selection of boys?

A: The advantage of using a table of randomly generated numbers to simulate the selection of boys is that it allows us to estimate the probability that we must select at least four boys to find one boy who commutes by bicycle. This is because the table of randomly generated numbers represents a random sample of the population, and we can use it to simulate the selection of boys.

Q: Can you explain the concept of probability in the context of this problem?

A: Yes, the concept of probability in the context of this problem is the chance of selecting a boy who commutes by bicycle. In this case, the probability is 0.4, which means that there is a 40% chance that a randomly selected boy commutes by bicycle.

Q: How did you estimate the probability that we must select at least four boys to find one boy who commutes by bicycle?

A: We estimated the probability by repeating the simulation many times and counting the number of times that we needed to select at least four boys to find one boy who commutes by bicycle. We then divided the number of times by the total number of simulations to get an estimate of the probability.

Q: What is the significance of the estimated probability?

A: The estimated probability that we must select at least four boys to find one boy who commutes by bicycle is significant because it gives us an idea of how many boys we need to select to find one who commutes by bicycle. In this case, we need to select at least four boys to find one who commutes by bicycle.

Q: Can you explain the concept of simulation in the context of this problem?

A: Yes, the concept of simulation in the context of this problem is the process of using a table of randomly generated numbers to simulate the selection of boys. We can use this simulation to estimate the probability that we must select at least four boys to find one boy who commutes by bicycle.

Q: How did you use the simulation to estimate the probability?

A: We used the simulation to estimate the probability by repeating the simulation many times and counting the number of times that we needed to select at least four boys to find one boy who commutes by bicycle. We then divided the number of times by the total number of simulations to get an estimate of the probability.