Suppose that only 25% of all drivers come to a complete stop at an intersection having flashing red lights
in all directions when no other cars are visible. What is the probability that, of 20 randomly chosen drivers
coming to an intersection under these conditions, at least 2 will come to a complete stop?
this is a binary problem ... s is stop , n is not stop
p(s) = .25 , p(n) = .75
(n + s)^20 = n^20 + 20 n^19 s + ... + 20 n s^19 + s^20
p(s ≥ 2) = 1 - [.75^20 + 20 (.75^19 * .25)]
74
To find the probability that at least 2 drivers will come to a complete stop out of 20 randomly chosen drivers, we need to calculate the probability of each possible outcome where 2 or more drivers come to a complete stop and add them together.
The probability of exactly 2 drivers coming to a complete stop can be calculated using the binomial probability formula:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
where:
- n is the total number of trials (20 drivers)
- k is the number of successful trials (2 drivers coming to a complete stop)
- p is the probability of success in a single trial (25% or 0.25)
Using this formula, we can calculate the probability of exactly 2 drivers coming to a complete stop.
P(X = 2) = (20 C 2) * 0.25^2 * (1 - 0.25)^(20 - 2)
Next, we calculate the probability of exactly 3 drivers coming to a complete stop.
P(X = 3) = (20 C 3) * 0.25^3 * (1 - 0.25)^(20 - 3)
We continue this process up to 20 drivers.
Finally, we add up all these probabilities to find the probability that at least 2 drivers come to a complete stop:
P(at least 2 drivers) = P(X = 2) + P(X = 3) + ... + P(X = 20)
To calculate the probability that at least 2 out of 20 randomly chosen drivers come to a complete stop at the intersection, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of exactly k successes
- C(n, k) is the combination of n items taken k at a time
- p is the probability of success (in this case, the probability that a driver comes to a complete stop)
- n is the number of trials (in this case, the number of drivers chosen)
In this case, we want to find the probability of at least 2 drivers coming to a complete stop. This means we need to calculate the probabilities for 2, 3, 4, ..., and up to 20 successful stops, and then sum them up.
P(X ≥ 2) = P(X = 2) + P(X = 3) + ... + P(X = 20)
First, we need to calculate P(X = k) for each value of k from 2 to 20:
P(X = k) = C(20, k) * 0.25^k * (1 - 0.25)^(20 - k)
Now we can substitute the values and calculate the individual probabilities for each value of k:
P(X = 2) = C(20, 2) * 0.25^2 * (1 - 0.25)^(20 - 2)
P(X = 3) = C(20, 3) * 0.25^3 * (1 - 0.25)^(20 - 3)
...
P(X = 20) = C(20, 20) * 0.25^20 * (1 - 0.25)^(20 - 20)
Finally, we add up all these probabilities to get the probability of at least 2 drivers coming to a complete stop:
P(X ≥ 2) = P(X = 2) + P(X = 3) + ... + P(X = 20)
By calculating these individual probabilities and adding them up, we can determine the probability that at least 2 out of 20 randomly chosen drivers come to a complete stop at the intersection when no other cars are visible.