Question

Mrs. Wheeler prepares a list of 434343 US presidents, 313131 of whom served in the military. Then 888 students each select a president at random (there can be repeats) for their civics presentations.
What is the probability that at least one of the students will select a president who did not serve in the military?

Answer 1

Answer:

Probability of atleast one of the students(888) selecting a president with no service in the military reaches 1.

Step-by-step explanation:

Total Presidents in list = 434343

Military men to be president = 313131

Probability of selecting President who were also military men =

p = 313131/434343 = 0.72

Probability of selecting President who were not military men =

q = 1 - p = 1 - 0.72 = 0.28

Now; no of students who make a choice = 888

no. of Choices made resulting in success = x : {0, 1, 2, 3,............., 888 }

GIVEN CASE:

P(f) = Probability of atleast one student selecting a president with no service in military

This case fails when no one selects a president with no service in military, let us call it P(f').

P(f) = 1-P(f')

Calculating P(f'):

Let us define:

Failure = selecting a president with service in military , p = 0.72

Success = selecting a president with no service in military , q = 0.28

Using Binomial Theorem:

we have this case when n students make selections and x of them are successful.

$P(x) = nCx * q^{x} * p^{n-x}$

In case of f' , n = 888 and x = 0

$P(0) = 888 C 0 * (0.28)^{0} * 0.72^{888-0}$

$= (888!/(888-0)!) * (1) * (0.72)^{888}\\= (0.72)^{888}\\= 2.047655e-127$

Hence, P(f') = 2.047655e-127 (reaches 0)

Now: P(f) = 1 - P(f')

P(f) = 1 - 2.047655e-127 = 1

Hence Probability of atleast one of the students(888) selecting a president with no service in the military is 1.