Question

In: Statistics and Probability

In a large introductory statistics lecture​ hall, the professor reports that 53​% of the students enrolled...

In a large introductory statistics lecture​ hall, the professor reports that 53​% of the students enrolled have never taken a calculus​ course, 28​% have taken only one semester of​ calculus, and the rest have taken two or more semesters of calculus. The professor randomly assigns students to groups of three to work on a project for the course. You are assigned to be part of a group. What is the probability that of your other two​ groupmates,

​a) neither has studied​ calculus?

​b) both have studied at least one semester of​ calculus?

​c) at least one has had more than one semester of​ calculus?

Solutions

Expert Solution

As the students are randomly assigned to each group , thus event of of taking calculus course is independent of each other.

P( never taken Calculus) =0.53

P( one semester of Calculus) = 0.28

P( two or more semester of Calculus) = 0.19

a) P( neither has studied Calculus) = P( both never taken Calculus)

As events are independent

P( neither has studied Calculus) = 0.53 *0.53 = 0.2809

b) P( both have studied at least one semester of Calculus)

= P( both have studied one semester of Calculus) + P( both have studied two or more semester of Calculus)

+ P( either of them have studied one semester of Calculus and another two or more semester of Calculus)

= 0.28 *0.28 + 0.19*0.19 +2 *0.28*0.19

= 0.2209

c) P( atleast one have studied more than one semester of Calculus)

= P( either of them have studied two or more semester of Calculus and another one not  )

+ P( both of them have studied two or more semester of Calculus )

= 2*0.19*(1-0.19 ) + 0.19*0.19

= 0.3439

Note : IF events A and B are independent

P( AB) =P( A ) *P( B)

Note : To find the following probabilities we multiplied by two , as the event can happen two ways

first person have studied one semester of Calculus and second two or more semester of Calculus

or

first person have studied wo or more semester of Calculus and second one semester of Calculus

P( either of them have studied one semester of Calculus and another two or more semester of Calculus)

= 0.19*0.28 +0.28 *0.19

Similarly

P( either of them have studied two or more semester of Calculus and another one not  )

orchestra answered 2 years ago
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