Question

65% of the people in Missouri pass the driver's test on the first attempt. A group of 7 people took the test. Which of the following equations correctly calculate the probability that at least 3 in the group pass their driver's tests in their first atempt? Select all that apply. Remember: 65%=0.65. Select all that apply: o P(X≥3)=P(X=3)+P(X=2)+P(X=1)+P(X=0)

o P(X≥3)=P(X=3)+P(X=4)+P(X=5)+P(X=6)+P(X=7)
o P(X≥3)=1−P(X=3)
o P(X≥3)=1−[P(X=2)+P(X=1)+P(X=0)]

Answer 1

To calculate the probability that at least 3 out of 7 people pass their driver's test on the first attempt, you can use the complement rule, which states:

P(X ≥ k) = 1 - P(X < k)

In this case, k represents the minimum number of people passing, which is 3 or more. So, you need to calculate the probability that fewer than 3 people pass, and then subtract that from 1.

Let's calculate it step by step:

P(X < 3) means the probability that fewer than 3 people pass on their first attempt. You can calculate this using the binomial probability formula:

P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)

Now, calculate each of these probabilities:

P(X = 0) = (0.35^7) * (1 - 0.35)^(7 - 0) * C(7, 0) = 0.00082236891 (Using the binomial probability formula)

P(X = 1) = (0.35^7) * (1 - 0.35)^(7 - 1) * C(7, 1) = 0.00765760794

P(X = 2) = (0.35^7) * (1 - 0.35)^(7 - 2) * C(7, 2) = 0.031763425

Now, add these probabilities together to find P(X < 3):

P(X < 3) = 0.00082236891 + 0.00765760794 + 0.031763425 ≈ 0.04024340185

Finally, use the complement rule to find P(X ≥ 3):

P(X ≥ 3) = 1 - P(X < 3) ≈ 1 - 0.04024340185 ≈ 0.95975659815

So, the probability that at least 3 people pass their driver's tests on the first attempt is approximately 0.9598 (rounded to four decimal places).

Therefore, the correct equation to calculate this probability is:

P(X ≥ 3) = 1 - P(X < 3)