Question

according to data from the unesco institute for statistics, globally, 30% of adolescents are proficient in reading and mathematics. two adolescent students are selected at random, one from a school in the united states, and one from a school in europe. knowing that the proficiency of these two students in reading and mathematics is independent of one another: (enter your answers below without rounding them) a) what is the probability that both students are proficient in reading and mathematics? 0.09 b) what is the probability that at least one of the students is proficient in reading and mathematics? 0.51 c) what is the probability that neither student is proficient in reading and mathematics? 0.49

Explanation:

Step1: Define probabilities

Let \(P(A)\) be the probability that a student is proficient. Given \(P(A)=0.3\), then the probability that a student is not proficient \(P(\overline{A}) = 1 - 0.3=0.7\)

Step2: Solve part (a)

Since the events are independent, the probability that both are proficient is \(P(A)\times P(A)\)

$$P(A_1\cap A_2)=0.3\times0.3 = 0.09$$

Step3: Solve part (b)

The probability that at least one is proficient is \(1 - P(\overline{A}_1\cap\overline{A}_2)\)

$$P(\overline{A}_1\cap\overline{A}_2)=0.7\times0.7 = 0.49$$

$$P(\text{at least one})=1 - 0.49=0.51$$

Step4: Solve part (c)

The probability that neither is proficient is \(P(\overline{A}_1\cap\overline{A}_2)\)

$$P(\overline{A}_1\cap\overline{A}_2)=0.7\times0.7 = 0.49$$

Answer:

a) \(0.09\)
b) \(0.51\)
c) \(0.49\)