Question 12.11E3: Spring Break According to a survey conducted by OnCampus Res......

a)  We want to determine the probability that exactly 3 of 5 college students selected at random will work during spring break. Therefore a college student working during spring break is considered a success. Thus, x = 3 and n = 5. The probability of success, p, is 20%, or 0.2. The probability of failure, q, is 1–0.2, or 0.8. Substituting these values into the binomial formula yields

P(x)  =  (_nC_x)p^xq^{n-x}
P(3)  =  (_5C_3)(0.2)^3(0.8)^{5-3}
=  10(0.2)^3(0.8)^2
=  10(0.008)(0.64)
=  0.0512

Thus, the probability that exactly 3 of 5 randomly selected college students will work during spring break is 0.0512.
b)  We want to find the probability that 4 of 4 college students selected at random will work during spring break. Thus, x = 4 and n = 4. We wish to find P(4).

P(x)  =  (_nC_x)p^xq^{n-x}
P(4)  =  (_4C_4)(0.2)^4(0.8)^{4-4}
=  1(0.2)^4(0.8)^0
=  10(0.0016)(1)
=  0.016

Thus, the probability that exactly 4 of 4 randomly selected college students will work during spring break is 0.016.