Answer to Question #163283 in Statistics and Probability for Dush

Question

It has been determined that 5% of drivers checked at a road stop show traces of alcohol and 10% of drivers checked do not wear seat belts. In addition, it has been observed that the two infractions are independent from one another. If an officer stops five drivers at random:

a. Calculate the probability that exactly three of the drivers have committed any one of the two offenses.

b. Calculate the probability that at least one of the drivers checked has committed at least one of the two

offenses.

Answer 1

P= 5 % = 0.05

p = 10 % = 0.10

n = 5

$P(X = x) = nC_x \times p^x \times (1 - p)^{n – x}$

$a. \; P(X= 3) = 5C_3 \times 0.05^3 \times (1-0.05)^{5-3} = 5C_3 \times 0.05^3 \times (0.95)^{2} \\ = 0.011 \\ b. \; P(X≥1) = ( 5C_1 \times 0.05^1 \times (0.95)^{5-1} ) + ( 5C_1 \times 0.10^1 \times (0.90)^{5-1} ) + ( 5C_1 \times 0.05^1 \times (0.95)^{5-1} ) \times ( 5C_1 \times 0.10^1 \times (0.90)^{5-1} ) \\ = 0.2262 + 0.4095 + (0.2262 \times 0.4095) \\ = 0.7283$