Question

Customers leaving a subway station can exit through any one of three gates. Assuming that any particular customer is equally likely to select any one of the three gates, find the probability that among a sample of 4 customers, they all exit through the same gate.

Answer 1

Answer: 0.0123

Step-by-step explanation:

Given : Customers leaving a subway station can exit through any one of three gates.

The probability of selecting any particular gate = $\dfrac{1}{3}$

Sample size : n= 4

Using binomial distribution,

$P(x)=^nC_xp^x(1-p)^{n-x}$, where P(x) is the probability of getting success inn x trials , n is the number of trails and  p is the probability of getting success in each trial.

Then , the probability that among a sample of 4 customers, they all exit through the same gate.

$P(4)=^4C_4(\dfrac{1}{3})^4(\dfrac{2}{3})^{0}=(1)(\dfrac{1}{3})^4=0.0123456790123\approx0.0123$

Hence, the  probability that among a sample of 4 customers, they all exit through the same gate = 0.0123