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A stoplight at the corner of Lincolnway and Duff Avenue is red 20% of the time. For
someone who drives through this stoplight 10 times a week, what is the
probability that they will stop at least three times?

a: .201
b: .771
c: .878
d: .322

My stats book wasn't very helpful in the teaching of this. I'm not really sure how to do it, so if you could teach me some EASIER methods, I would greatly appreciate it!

You can use a binomial probability table (which is much easier) or do this by hand with the following formula:

P(x) = (nCx)(p^x)[q^(n-x)]

x = 0, 1, 2
n = 10
p = .2
q = 1 - p = 1 - .2 = .8

Calculate P(0), P(1), and P(2).

Add P(0), P(1), and P(2) probabilities together, than subtract this value from 1. You should have your answer!

I hope this helps.

Got it! Thanks!!

To solve this problem, we will use the binomial probability formula:

P(X ≥ k) = 1 - P(X < k)

where P(X < k) is the cumulative probability that X takes on values less than k.

In this case, X represents the number of times the stoplight is red out of the 10 times someone drives through the stoplight in a week. The probability of the stoplight being red is 20%, so the probability of it being green is 80%.

Now let's calculate the probability that they will stop at least three times:

P(X ≥ 3) = 1 - P(X < 3)

To find P(X < 3), we need to calculate the probabilities for X = 0, 1, and 2, and then sum them up.

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

P(X = 0) = (0.8)^10 (since the stoplight is green all 10 times)
P(X = 1) = 10C1 * (0.2)^1 * (0.8)^9 (since the stoplight is red once and green nine times)
P(X = 2) = 10C2 * (0.2)^2 * (0.8)^8 (since the stoplight is red twice and green eight times)

Now let's calculate these probabilities:

P(X = 0) = (0.8)^10 ≈ 0.1074
P(X = 1) = 10C1 * (0.2)^1 * (0.8)^9 ≈ 0.2684
P(X = 2) = 10C2 * (0.2)^2 * (0.8)^8 ≈ 0.3010

Now let's calculate P(X < 3):

P(X < 3) = 0.1074 + 0.2684 + 0.3010 ≈ 0.6768

Finally, we can calculate the probability of stopping at least three times:

P(X ≥ 3) = 1 - P(X < 3) ≈ 1 - 0.6768 ≈ 0.3232

So, the correct answer is d: 0.3232.