Question

"You ask students to identify European countries on a map. On average students will identify 10% of the countries correctly. A similar test for US states indicates a 45% success rate. Asking students to identify 7 European countries and 3 US states, what is the chance that students will correctly identify at least 50% of the countries/states?"

"You have a set of 100 batteries. 13 of these batteries are defective. Testing 25 batteries, what is the probability to find 3 or more of the 25 batteries failing?"

Answer 1

Answer:

part 1:

Given,

To give the required probability

P(students who identify countries correctly) = 0.10

P(c) = 0.10

P(students who identify states correctly) = 0.45

P(s) = 0.45

Now consider,

P(c ∩ s) = P(c)*P(s)

substitute values given

= 0.10*0.45

P(c ∩ s) = 0.045

Now the probability to identify at least 50% of countries/states correctly = P(c ∩ s) / 50%

= 0.045 / 0.5

Required probability = 0.09

part 2:

To determine the probability that to find 3 or more of 25 batteries failing

Given,

13 out of 100 batteries are defective

P(defective) = 13/100

= 0.13

q = 1 - p

= 1 - 0.13

q = 0.87

let us assume 25 batteries selected at random i.e., n = 25

P(X = x) = nCx * p^x * q^(n-x)

substitute values

= 25Cx * (13/100)^x * (87/100)^25-x

Now to give probability of selecting at least 3 defective = P(x >= 3)

= 1 - P(x < 3)

= 1 - [P(x = 2) + P(x = 1) + P(x = 0)]

= 1 - [25C2*0.13^2*0.87^23 + 25C1*0.13^1*0.87^24 + 25C0*0.13^0*0.87^25]

= 1 - [0.2060 + 0.1149 + 0.0308]

= 1 - 0.3517

Required probability = 0.6483